We consider Schrödinger equations with competing nonlinearit ies in spatial dimensions up to three\, for which global existence holds ( i.e. no finite-time blow-up). A typical example is the case of the (focusi ng-defocusing) cubic-quintic NLS. \;We recall the notions of energy mi nimizing versus action minimizing ground states and show that\, in general \, the two must be considered as nonequivalent. The question of long-time behavior of solutions\, in particular the problem of ground-state (in-)sta bility will be discussed using analytical results and numerical simulation s.

DTEND;TZID=Europe/Zurich:20240522T151500 END:VEVENT BEGIN:VEVENT UID:news1688@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240508T175114 DTSTART;TZID=Europe/Zurich:20240515T151500 SUMMARY:Seminar Analysis and Mathematical Physics: Min Jun Jo (Duke Univers ity) DESCRIPTION:We prove the instantaneous cusp formation from a single corner of the vortex patch solutions. This positively settles the conjecture give n by Cohen-Danchin in Multiscale approximation of vortex patches\, SIAM J . Appl. Math. 60 (2000)\, no. 2\, 477–502. X-ALT-DESC:We prove the instantaneous cusp formation from a single corne
r of the vortex patch solutions. This positively settles the conjecture gi
ven by Cohen-Danchin in \;*Multiscale approximation of vortex patch
es\, SIAM J. Appl. Math. 60 (2000)\, no. 2\, 477–502.*

This talk concerns critical points $u$ of polyconvex energies of the form $f(X) = g(det(X))$\, where $g$ is (uniformly) convex. It is n ot hard to see that\, if $u$ is smooth\, then $\\det(Du)$ is constant. I w ill show that the same result holds for Lipschitz critical points $u$ in t he plane. I will also discuss how to obtain rigidity for approximate solut ions. This is a joint work with A. Guerra.

DTEND;TZID=Europe/Zurich:20240515T150000 END:VEVENT BEGIN:VEVENT UID:news1689@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240503T101127 DTSTART;TZID=Europe/Zurich:20240508T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Christoph Kehle (ETH Zü rich) DESCRIPTION:Extremal black holes are special types of black holes which hav e exactly zero temperature. I will present a proof that extremal black hol es form in finite time in gravitational collapse of charged matter. In par ticular\, this construction provides a definitive disproof of the “third law” of black hole thermodynamics. I will also present a recent result which shows that extremal black holes arise on the black hole formation th reshold in the moduli space of gravitational collapse. This gives rise to a new conjectural picture of “extremal critical collapse.” This is joi nt work with Ryan Unger (Princeton). X-ALT-DESC:Extremal black holes are special types of black holes which h ave exactly zero temperature. I will present a proof that extremal black h oles form in finite time in gravitational collapse of charged matter. In p articular\, this construction provides a definitive disproof of the “thi rd law” of black hole thermodynamics. I will also present a recent resul t which shows that extremal black holes arise on the black hole formation threshold in the moduli space of gravitational collapse. This gives rise t o a new conjectural picture of “extremal critical collapse.” This is j oint work with Ryan Unger (Princeton).

DTEND;TZID=Europe/Zurich:20240508T160000 END:VEVENT BEGIN:VEVENT UID:news1666@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240422T090328 DTSTART;TZID=Europe/Zurich:20240424T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Louise Gassot (IRMAR\, R ennes) DESCRIPTION:We focus on the Benjamin-Ono equation on the line with a small dispersion parameter. The goal of this talk is to precisely describe the s olution at all times when the dispersion parameter is small enough. This s olution may exhibit locally rapid oscillations\, which are a manifestation of a dispersive shock. The description involves the multivalued solution of the underlying Burgers equation\, obtained by using the method of chara cteristics. This work is in collaboration with Elliot Blackstone\, Patrick Gérard\, and Peter Miller. X-ALT-DESC:We focus on the Benjamin-Ono equation on the line with a smal l dispersion parameter. The goal of this talk is to precisely describe the solution at all times when the dispersion parameter is small enough. This solution may exhibit locally rapid oscillations\, which are a manifestati on of a dispersive shock. The description involves the multivalued solutio n of the underlying Burgers equation\, obtained by using the method of cha racteristics. This work is in collaboration with Elliot Blackstone\, Patri ck Gérard\, and Peter Miller.

DTEND;TZID=Europe/Zurich:20240424T151500 END:VEVENT BEGIN:VEVENT UID:news1638@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240328T104112 DTSTART;TZID=Europe/Zurich:20240417T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Anuj Kumar (UC Berkeley) DESCRIPTION:We construct nonunique solutions of the transport equation in t he class $L^\\infty$ in time and $L^r$ in space for divergence free Sobole v vector fields $W^{1\, p}$. We achieve this by introducing two novel idea s: (1) In the construction\, we interweave the scaled copies of the vector field itself. (2) Asynchronous translation of cubes\, which makes the con struction heterogeneous in space. These new ideas allow us to prove nonuni queness in the range of exponents beyond what is available using the metho d of convex integration and sharply matchwith the range of uniqueness of s olutions from Bruè\, Colombo\, De Lellis ’21. X-ALT-DESC:We construct nonunique solutions of the transport equation in the class $L^\\infty$ in time and $L^r$ in space for divergence free Sobo lev vector fields $W^{1\, p}$. We achieve this by introducing two novel id eas: (1) In the construction\, we interweave the scaled copies of the vect or field itself. (2) Asynchronous translation of cubes\, which makes the c onstruction heterogeneous in space. These new ideas allow us to prove nonu niqueness in the range of exponents beyond what is available using the met hod of convex integration and sharply matchwith the range of uniqueness of solutions from Bruè\, Colombo\, De Lellis ’21. \;

DTEND;TZID=Europe/Zurich:20240417T160000 END:VEVENT BEGIN:VEVENT UID:news1619@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240328T104000 DTSTART;TZID=Europe/Zurich:20240410T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Roman Shvydkoy (Universi ty of Illinois at Chicago) DESCRIPTION:The classical Kolmogorov-41 theory of turbulence is based on a set of pivotal assumptions on scaling and energy dissipation for solutio ns satisfying incompressible fluid models. In the early 80's experimental evidence emerged that pointed to departure from the K41 predictions\, whic h was attributed to the phenomenon of statistical intermittency. In this talk we give an overview of the classical results in the subject\, relati onship of intermittency to the problem of global well-posedness of the 3D Navier-Stokes system\, and discuss a new approach developed jointly with A . Cheskidov on how to measure and study intermittency from a rigorous pers pective. At the center of our discussion will be a new interpretation of an intermittent signal described by volumetric properties of the filter ed field. It provides\, in particular\, a systematic approach to the Fr isch-Parisi multifractal formalism\, and recasts intermittency from the po int of view of information theory. X-ALT-DESC:The classical Kolmogorov-41 theory of turbulence is based on a set of \; pivotal assumptions on scaling and energy dissipation for solutions satisfying incompressible fluid models. In the early 80's experi mental evidence emerged that pointed to departure from the K41 predictions \, which was attributed to the phenomenon of statistical intermittency.&nb sp\; In this talk we give an overview of the classical results in the subj ect\, relationship of intermittency to the problem of global well-posednes s of the 3D Navier-Stokes system\, and discuss a new approach developed jo intly with A. Cheskidov on how to measure and study intermittency from a r igorous perspective. \; At the center of our discussion will be a new interpretation of an intermittent signal \;described \;by volumetr ic properties of the filtered \;field. \; It provides\, in particu lar\, a systematic approach to the Frisch-Parisi multifractal formalism\, and recasts intermittency from the point of view of information theory.&nb sp\;

DTEND;TZID=Europe/Zurich:20240410T160000 END:VEVENT BEGIN:VEVENT UID:news1648@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240323T092733 DTSTART;TZID=Europe/Zurich:20240403T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Vikram Giri (ETH Zürich ) DESCRIPTION:A common issue in convex integration methods (going back to Nas h) is the presence of unwanted high-high frequency interactions. These iss ues can prevent the methods from producing solutions with the optimum "Ons ager" regularity to a given system of PDEs. We will discuss these issues i n the setting of the 2D Euler equations and then discuss a linear Newton i teration designed to get rid of these unwanted interactions in this settin g. We will conclude by discussing applications to other PDEs. This is base d on joint works with Răzvan-Octavian Radu and Mimi Dai. X-ALT-DESC:A common issue in convex integration methods (going back to N ash) is the presence of unwanted high-high frequency interactions. These i ssues can prevent the methods from producing solutions with the optimum "O nsager" regularity to a given system of PDEs. We will discuss these issues in the setting of the 2D Euler equations and then discuss a linear Newton iteration designed to get rid of these unwanted interactions in this sett ing. We will conclude by discussing applications to other PDEs. This is ba sed on joint works with Răzvan-Octavian Radu and Mimi Dai.

DTEND;TZID=Europe/Zurich:20240403T160000 END:VEVENT BEGIN:VEVENT UID:news1645@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240325T134941 DTSTART;TZID=Europe/Zurich:20240327T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Marcello Porta (SISSA) DESCRIPTION:I will discuss the dynamics of many-body Fermi gases\, in the m ean-field regime. I will consider a class of initial data which are close enough to quasi-free states\, with a non-zero pairing matrix. Assuming a suitable semiclassical structure for the initial datum\, expected to hold at low enough energy and that we can establish for translation-invariant states\, I will present a theorem that shows that the many-body evolution of the system can be well approximated by the Hartree-Fock-Bogoliubov equ ation\, a non-linear effective evolution equation describing the coupled dynamics of the reduced one-particle density matrix and of the pairing ma trix. Joint work with Stefano Marcantoni (Nice) and Julien Sabin (Rennes). X-ALT-DESC:I will discuss the dynamics of many-body Fermi gases\, in the mean-field regime. I will consider a class of initial data which are \;close enough to quasi-free states\, with a non-zero pairing matrix. Assu ming a suitable semiclassical structure for the initial datum\, \;expe cted to hold at low enough energy and that we can establish for translatio n-invariant states\, I will present a theorem that shows that \;the ma ny-body evolution of the system can be well approximated by the Hartree-Fo ck-Bogoliubov equation\, a non-linear effective \;evolution equation d escribing the coupled dynamics of the reduced one-particle density matrix and of \;the pairing matrix. Joint work with Stefano Marcantoni (Nice) and Julien Sabin (Rennes).

DTEND;TZID=Europe/Zurich:20240327T153000 END:VEVENT BEGIN:VEVENT UID:news1641@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240205T090917 DTSTART;TZID=Europe/Zurich:20240320T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Norbert J. Mauser (WPI and MMM c/o Univ. Wien) DESCRIPTION:The Pauli-Poisswell equation models fast moving charges in\\r\\ nsemiclassical semi-relativistic quantum dynamics. It is at the\\r\\ncente r of a hierarchy of models from the Dirac-Maxwell equation to\\r\\nthe Eul er-Poisson equation that are linked by asymptotic analysis of\\r\\nsmall p arameters such as Planck constant or inverse speed of light.\\r\\nWe discu ss the models and their application in plasma and\\r\\naccelerator physics as well as the many mathematical problems they\\r\\npose. X-ALT-DESC:The Pauli-Poisswell equation models fast moving charges in

\nsemiclassical semi-relativistic quantum dynamics. It is at the

\ncenter of a hierarchy of models from the Dirac-Maxwell equation to

\ nthe Euler-Poisson equation that are linked by asymptotic analysis of\n

small parameters such as Planck constant or inverse speed of light.

\nWe discuss the models and their application in plasma and

\naccelerator physics as well as the many mathematical problems they

\n< p>pose. DTEND;TZID=Europe/Zurich:20240320T161500 END:VEVENT BEGIN:VEVENT UID:news1639@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240201T155918 DTSTART;TZID=Europe/Zurich:20240306T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Klaus Widmayer (Universi tät Zürich) DESCRIPTION:While "Landau damping" is regarded as an important effect in th e dynamics of hot\, collisionless plasmas\, its mathematical understanding is still in its infancy. This talk presents a recent nonlinear stability result in this context. Starting with a discussion of stabilizing mechanis ms in the linearized Vlasov-Poisson equations near a class of homogeneous equilibria on R^3\, we will see how both oscillatory and damping effects a rise\, and sketch how these mechanisms imply a nonlinear stability result in the specific setting of the Poisson equilibrium. This is based on joint work with A. Ionescu\, B. Pausader and X. Wang. X-ALT-DESC:While "Landau damping" is regarded as an important effect in the dynamics of hot\, collisionless plasmas\, its mathematical understandi ng is still in its infancy. This talk presents a recent nonlinear stabilit y result in this context. Starting with a discussion of stabilizing mechan isms in the linearized Vlasov-Poisson equations near a class of homogeneou s equilibria on R^3\, we will see how both oscillatory and damping effects arise\, and sketch how these mechanisms imply a nonlinear stability resul t in the specific setting of the Poisson equilibrium. This is based on joi nt work with A. Ionescu\, B. Pausader and X. Wang.

DTEND;TZID=Europe/Zurich:20240306T160000 END:VEVENT BEGIN:VEVENT UID:news1611@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231107T040532 DTSTART;TZID=Europe/Zurich:20231213T151500 SUMMARY:Seminar Analysis and Mathematical Physics: Theodore Drivas (Stony B rook University) DESCRIPTION:We will discuss aspects of the global picture of 2D fluids. Ste ady states\, deterioration of regularity for time dependent solutions as w ell as for the Lagrangian flowmap\, as well as conjectural pictures about the weak-* attractor and generic behavior by Shnirelman and Sverak. \\r\\ nNotice the special time! X-ALT-DESC:We will discuss aspects of the global picture of 2D fluids. S teady states\, deterioration of regularity for time dependent solutions as well as for the Lagrangian flowmap\, as well as conjectural pictures abou t the weak-* attractor and generic behavior by Shnirelman and Sverak. \;

\n**Notice the special time!**

A vortex ring is a solution of the axisymmetric Euler equatio ns consisting of some torus of concentrated vorticity. Motivated by the ap pearance of Vortex rings as bubble rings\, we study vortex rings with surf ace tension at the interface. We show the existence of traveling wave solu tions. In particular\, our construction also justifies the existence of so -called hollow vortex rings\, where the vorticity is a measure concentrate d on the interface.

DTEND;TZID=Europe/Zurich:20231206T160000 END:VEVENT BEGIN:VEVENT UID:news1580@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231114T180149 DTSTART;TZID=Europe/Zurich:20231129T134500 SUMMARY:Seminar Analysis and Mathematical Physics: Sergio Simonella (Univer sity of Roma La Sapienza) DESCRIPTION:Boltzmann equation\, hard sphere systems and their small and la rge deviations X-ALT-DESC:Boltzmann equation\, hard sphere systems and their small and lar ge deviations DTEND;TZID=Europe/Zurich:20230929T163000 END:VEVENT BEGIN:VEVENT UID:news1614@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231114T180522 DTSTART;TZID=Europe/Zurich:20231129T134500 SUMMARY:Seminar Analysis and Mathematical Physics: Sergio Simonella (Univer sity of Roma La Sapienza) DESCRIPTION:https://nccr-swissmap.ch/news-and-events/news/next-kinetic-theo ry-seminar-29th-nov-prof-sergio-simonella-university-roma-la-sapienza X-ALT-DESC:https://nccr-swissmap.ch/news-and-events/news/next-kinetic-th eory-seminar-29th-nov-prof-sergio-simonella-university-roma-la-sapienza

DTEND;TZID=Europe/Zurich:20231129T163000 END:VEVENT BEGIN:VEVENT UID:news1577@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231029T223020 DTSTART;TZID=Europe/Zurich:20231122T140000 SUMMARY:Seminar Analysis and Mathematical Physics: Jinyeop Lee (LMU Munich) DESCRIPTION:The study of the Schrödinger equation in dimension one with a nonlinear point interaction has been the focus of research over the past f ew decades. In this seminar\, we talk about a work on deriving this partia l differential equation as the effective dynamics of N identical bosons in one dimension. We assume introducing a tiny impurity located at the origi n and considering that the interaction between every pair of bosons is med iated by the impurity through a three-body interaction. Moreover\, by assu ming short-range scaling and choosing an initial fully condensed state\, w e prove convergence of one-particle density operators in the trace-class t opology. This is the first derivation of the so-called nonlinear delta mod el. This research is a collaborative work with Prof. Riccardo Adami. X-ALT-DESC:The study of the Schrödinger equation in dimension one with
a nonlinear point interaction has been the focus of research over the past
few decades. In this seminar\, we talk about a work on deriving this part
ial differential equation as the effective dynamics of N identical bosons
in one dimension.

We assume introducing a tiny impurity located at t
he origin and considering that the interaction between every pair of boson
s is mediated by the impurity through a three-body interaction. Moreover\,
by assuming short-range scaling and choosing an initial fully condensed s
tate\, we prove convergence of one-particle density operators in the trace
-class topology. This is the first derivation of the so-called nonlinear d
elta model. This research is a collaborative work with Prof. Riccardo Adam
i.

In this talk we consider the long-time behavior of solutions to the two dimensional non-homogeneous Euler equations under the Boussines q approximation posed on a periodic channel. We prove inviscid damping &nb sp\;for the linearized equations around the stably stratified Couette flow using stationary-phase methods of oscillatory integrals. We discuss how t hese oscillatory integrals arise\, \;what are the main regularity req uirements to carry out the stationary-phase arguments\, and how to achieve such regularities.

DTEND;TZID=Europe/Zurich:20231115T160000 END:VEVENT BEGIN:VEVENT UID:news1602@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231027T193133 DTSTART;TZID=Europe/Zurich:20231108T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Gabriele Bocchi (Univers ità degli Studi di Roma Tor Vergata) DESCRIPTION:We analyze an optimal transport problem with additional entropi c cost evaluated along curves in the Wasserstein space which join two prob ability measures m_0 and m_1. The effect of the additional entropy functio nal results into an elliptic regularization for the (so-called) Kantorovic h potentials of the dual problem.\\r\\nAssuming the initial and terminal m easures to have densities\, we prove that the optimal curve remains positi ve and locally bounded in time. We focus on the case that the transport pr oblem is set on a compact Riemannian manifold with Ricci curvature bounded below.\\r\\nThe approach follows ideas introduced by P.L. Lions in the th eory of mean-field games about optimization problems with penalizing conge stion terms. Crucial steps of our strategy include displacement convexity properties in the Eulerian approach and the analysis of distributional sub solutions to Hamilton-Jacobi equations.\\r\\nThe result provides a smooth approximation of Wasserstein-2 geodesics. X-ALT-DESC:We analyze an optimal transport problem with additional entro pic cost evaluated along curves in the Wasserstein space which join two pr obability measures m_0 and m_1. The effect of the additional entropy funct ional results into an elliptic regularization for the (so-called) Kantorov ich potentials of the dual problem.

\nAssuming the initial and termi nal measures to have densities\, we prove that the optimal curve remains p ositive and locally bounded in time. We focus on the case that the transpo rt problem is set on a compact Riemannian manifold with Ricci curvature bo unded below.

\nThe approach follows ideas introduced by P.L. Lions i n the theory of mean-field games about optimization problems with penalizi ng congestion terms. Crucial steps of our strategy include displacement co nvexity properties in the Eulerian approach and the analysis of distributi onal subsolutions to Hamilton-Jacobi equations.

\nThe result provide s a smooth approximation of Wasserstein-2 geodesics.

DTEND;TZID=Europe/Zurich:20231108T160000 END:VEVENT BEGIN:VEVENT UID:news1594@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231017T092609 DTSTART;TZID=Europe/Zurich:20231025T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Ruoci Sun (Georgia Tech) DESCRIPTION:The intertwined derivative Schrödinger system of Calogero-Mose r type.\\r\\n(Online Seminar via Zoom). X-ALT-DESC:The intertwined derivative Schrödinger system of Calogero-Moser type.\n(Online Seminar via Zoom). DTEND;TZID=Europe/Zurich:20231025T031500 END:VEVENT BEGIN:VEVENT UID:news1559@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230919T123033 DTSTART;TZID=Europe/Zurich:20230927T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Bian Wu (MPI Leipzig) DESCRIPTION:I will talk about Rayleigh-Taylor instability for two miscible\ , incompressible\, inviscid fluids. Scale-invariant estimates for the size of the mixing zone and coarsening of internal structures in the fully non linear regime are established. These bounds provide optimal scaling laws a nd reveal the strong role of dissipation in slowing down mixing. This is a joint work with Konstantin Kalinin\, Govind Menon. X-ALT-DESC:I will talk about Rayleigh-Taylor instability for two miscibl e\, incompressible\, inviscid fluids. Scale-invariant estimates for the si ze of the mixing zone and coarsening of internal structures in the fully n onlinear regime are established. These bounds provide optimal scaling laws and reveal the strong role of dissipation in slowing down mixing. This is a joint work with Konstantin Kalinin\, Govind Menon.

DTEND;TZID=Europe/Zurich:20230927T160000 END:VEVENT BEGIN:VEVENT UID:news1547@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230904T075228 DTSTART;TZID=Europe/Zurich:20230920T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Chiara Boccato (Universi ty of Milano) DESCRIPTION:The interacting Bose gas is a system in quantum statistical mec hanics where a collective behavior emerges from the underlying many-body t heory\, posing interesting challenges to its rigorous mathematical descrip tion.\\r\\nWhile at temperature close to zero we have precise information on the ground state energy and the low-lying spectrum of excitations (at l east in certain scaling limits)\, much less is known close to the critical point. In this talk I will discuss how thermal excitations can be describ ed by Bogoliubov theory\, allowing us to estimate the free energy of the B ose gas in the Gross-Pitaevskii regime.\\r\\n\\r\\nThis is joint work with A. Deuchert and D. Stocker. X-ALT-DESC:The interacting Bose gas is a system in quantum statistical m echanics where a collective behavior emerges from the underlying many-body theory\, posing interesting challenges to its rigorous mathematical descr iption.

\nWhile at temperature close to zero we have precise informa tion on the ground state energy and the low-lying spectrum of excitations (at least in certain scaling limits)\, much less is known close to the cri tical point. In this talk I will discuss how thermal excitations can be de scribed by Bogoliubov theory\, allowing us to estimate the free energy of the Bose gas in the Gross-Pitaevskii regime.

\n\nThis is joint work with A. Deuchert and D. Stocker.

DTEND;TZID=Europe/Zurich:20230920T153000 END:VEVENT BEGIN:VEVENT UID:news1524@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230525T104748 DTSTART;TZID=Europe/Zurich:20230607T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Luis Martinez Zoroa (ICM AT Madrid) DESCRIPTION:The Surface quasi-geostrophic (SQG) equation is an important ac tive scalar model\, both due to its shared properties with 3D Euler as wel l as for its applications to model certain atmospherical phenomena. It has already been established that instantaneous loss of regularity can occur in the inviscid case in certain Sobolev spaces\, but it is unclear at what point the diffusion prevents this phenomenon from happening. In this talk I will discuss the behaviour when there is some super-critical fractional diffusion. X-ALT-DESC:The Surface quasi-geostrophic (SQG) equation is an important active scalar model\, both due to its shared properties with 3D Euler as w ell as for its applications to model certain atmospherical phenomena. It h as already been established that instantaneous loss of regularity can occu r in the inviscid case in certain Sobolev spaces\, but it is unclear at wh at point the diffusion prevents this phenomenon from happening. In this ta lk I will discuss the behaviour when there is some super-critical fraction al diffusion.

DTEND;TZID=Europe/Zurich:20230607T160000 END:VEVENT BEGIN:VEVENT UID:news1514@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230516T094922 DTSTART;TZID=Europe/Zurich:20230524T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Thérèse Moerschell (EP FL) DESCRIPTION:The advection-diffusion equation is known to have unique soluti ons for any vector field that is L^2 in time and in space. But what happen s when we have slightly less than square integrability? In this talk we wi ll explore two examples of vector fields in L^p(0\,T\;L^q(\\T^d)) made of shear flows that prove the non-uniqueness of solutions whenever we have p< 2 or q<2. We will first show that they give different solutions to the adv ection equation and then use the Feynman-Kac formula to show that diffusio n has little effect if our parameters are well-tuned. This is part of my M aster's thesis\, supervised by Massimo Sorella and Maria Colombo. X-ALT-DESC:The advection-diffusion equation is known to have unique solu
tions for any vector field that is L^2 in time and in space. But what happ
ens when we have slightly less than square integrability? In this talk we
will explore two examples of vector fields in L^p(0\,T\;L^q(\\T^d)) made o
f shear flows that prove the non-uniqueness of solutions whenever we have
p<\;2 or q<\;2. We will first show that they give different solutions
to the advection equation and then use the Feynman-Kac formula to show tha
t diffusion has little effect if our parameters are well-tuned.

This
is part of my Master's thesis\, supervised by Massimo Sorella and Maria C
olombo.

The phenomenon of anomalous dissipation in turbulence predict s the existence of solutions to the incompressible Euler equations that en joy regularity consistent with Kolmogorov’s 4/5 law and satisfy a local energy inequality. The "strong Onsager conjecture" asserts that such solut ions do indeed exist. In this talk\, we will discuss the background and mo tivation behind the strong Onsager conjecture. \; In addition\, we out line a construction of solutions with regularity (nearly) consistent with the 4/5 law\, thereby proving the conjecture in the natural L^3 scale of B esov spaces. \; This is based on joint work with Hyunju Kwon and Vikra m Giri.

DTEND;TZID=Europe/Zurich:20230517T160000 END:VEVENT BEGIN:VEVENT UID:news1483@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230418T150401 DTSTART;TZID=Europe/Zurich:20230426T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Ángel Castro (ICMAT\, M adrid) DESCRIPTION:In this talk we will consider the existence of traveling waves arbitrarily close to shear flows for the 2D incompressible Euler equat ions. In particular we shall present some results concerning the existence of such solutions near the Couette\, Taylor-Couette and the Poiseuille f lows. In the first part of the talk we will introduce the problem and revi ew some well known results on this topic. In the second one some of the ideas behind the construction of our traveling waves will be sketched. X-ALT-DESC:In this talk we will consider the existence of traveling wave s arbitrarily close to \; shear flows for \; the 2D incompressible Euler equations. In particular we shall present some results concerning t he existence of such solutions near the Couette\, Taylor-Couette and the P oiseuille \;flows. In the first part of the talk we will introduce the problem and review some well known results on this topic. In the \; s econd one some of the ideas behind the construction of our traveling waves \; will be sketched.

DTEND;TZID=Europe/Zurich:20230426T160000 END:VEVENT BEGIN:VEVENT UID:news1475@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230413T104028 DTSTART;TZID=Europe/Zurich:20230419T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Stefano Spirito (Univers ità degli Studi dell'Aquila) DESCRIPTION:I will review the existence and uniqueness theory of a model fo r viscoelastic materials of Kelvin-Voigt type with large strain. In partic ular\, I will first review the existence theory in L2\, and then show that also propagation of H1-regularity for the deformation gradient of weak so lutions in two and three dimensions holds. Moreover\, in two dimensions it is also possible to prove uniqueness of weak solutions. Additional pro pagation of higher regularity can be obtained\, leading to a global in tim e existence of smooth solutions. Joint work with K. Koumatos (U. of Sussex )\, C. Lattanzio (UnivAQ) and A. Tzavaras (KAUST). X-ALT-DESC:I will review the existence and uniqueness theory of a model for viscoelastic materials of Kelvin-Voigt type with large strain. In part icular\, I will first review the existence theory in L2\, and then show th at also propagation of H1-regularity for the deformation gradient of weak solutions in two and three dimensions holds. \; \;Moreover\, in tw o dimensions it is also possible to prove uniqueness of weak solutions. Ad ditional propagation of higher regularity can be obtained\, leading to a g lobal in time existence of smooth solutions. Joint work with K. Koumatos ( U. of Sussex)\, C. Lattanzio (UnivAQ) and A. Tzavaras (KAUST).

DTEND;TZID=Europe/Zurich:20230419T160000 END:VEVENT BEGIN:VEVENT UID:news1467@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230125T171400 DTSTART;TZID=Europe/Zurich:20230322T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Mickaël Latocca (Univer sité d'Évry) DESCRIPTION:In an incompressible fluid\, the pressure is governed by the el liptic equation $-\\Delta p = \\div \\div u \\otimes u$ and a Neuman-type boundary condition\, where $u$ stands for the divergence-free velocity vec tor field. The main goal of this talk is to explain why one expects that $p$ has Double Hölder regularity (with respect to that of $u$) and how on e can rigourously prove such a fact in a bounded domain. The results prese nted in this talk were obtained in collaboation with Luigi De Rosa (Basel) and Giorgio Stefani (SISSA). X-ALT-DESC:In an incompressible fluid\, the pressure is governed by the elliptic equation $-\\Delta p = \\div \\div u \\otimes u$ and a Neuman-typ e boundary condition\, where $u$ stands for the divergence-free velocity v ector field. \;The main goal of this talk is to explain why one expect s that $p$ has Double Hölder regularity (with respect to that of $u$) and how one can rigourously prove such a fact in a bounded domain. The result s presented in this talk were obtained in collaboation with Luigi De Rosa (Basel) and Giorgio Stefani (SISSA). \;

DTEND;TZID=Europe/Zurich:20230322T160000 END:VEVENT BEGIN:VEVENT UID:news1466@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230305T211240 DTSTART;TZID=Europe/Zurich:20230315T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Federico Cacciafesta (Un iversity of Padova) DESCRIPTION:The Dirac equation is one of the fundamental equations in relat ivistic quantum mechanics\, widely used in a large number of applications from physics to quantum chemistry. The aim of this talk will be to discuss some recent results\, together with a number of open questions\, concerni ng the dynamics for this model: after briefly reviewing the main propertie s of the Dirac operator and providing some background and motivations from the theory of linear dispersive PDEs\, we shall focus in particular on th e cases of the Dirac-Coulomb equation and of the Dirac equation on non fla t manifolds\, showing how some linear estimates (in particular\, Strichart z estimates) can be obtained by exploiting various properties of the opera tor. X-ALT-DESC:The Dirac equation is one of the fundamental equations in rel ativistic quantum mechanics\, widely used in a large number of application s from physics to quantum chemistry. The aim of this talk will be to discu ss some recent results\, together with a number of open questions\, concer ning the dynamics for this model: after briefly reviewing the main propert ies of the Dirac operator and providing some background and motivations fr om the theory of linear dispersive PDEs\, we shall focus in particular on the cases of the Dirac-Coulomb equation and of the Dirac equation on non f lat manifolds\, showing how some linear estimates (in particular\, Stricha rtz estimates) can be obtained by exploiting various properties of the ope rator.

DTEND;TZID=Europe/Zurich:20230315T161500 END:VEVENT BEGIN:VEVENT UID:news1465@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230119T132638 DTSTART;TZID=Europe/Zurich:20230308T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Maria Ahrend (Uni Basel) DESCRIPTION:Maria Ahrend will defend her PhD thesis on fractional Liouville equations and Calogero-Moser NLS. X-ALT-DESC:Maria Ahrend will defend her PhD thesis on fractional Liouvil le equations and Calogero-Moser NLS.

DTEND;TZID=Europe/Zurich:20230309T160000 END:VEVENT BEGIN:VEVENT UID:news1476@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230304T163156 DTSTART;TZID=Europe/Zurich:20230307T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Zineb Hassainia (New Yor k University Abu Dhabi) [special time!] DESCRIPTION:In this talk\, I will discuss a recent result concerning the co nstruction of quasi-periodic vortex patch solutions with one hole for th e 2D-Euler equations. These structures exist close to any annulus provided that its modulus belongs to a Cantor set with almost full Lebesgue measur e. The proof is based on a KAM reducibility scheme and a Nash-Moser iterative scheme. This is a joint work with Taoufik Hmidi and Emeric Roull ey. X-ALT-DESC:In this talk\, I will discuss a recent result concerning the construction of quasi-periodic vortex patch solutions \; with one hole for the 2D-Euler equations. These structures exist close to any annulus p rovided that its modulus belongs to a Cantor set with almost full Lebesgue measure. The proof is based on \; a KAM reducibility scheme \; an d a \; Nash-Moser iterative scheme. This is a joint work with Taoufik Hmidi and Emeric Roulley.

DTEND;TZID=Europe/Zurich:20230307T160000 END:VEVENT BEGIN:VEVENT UID:news1470@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230131T153947 DTSTART;TZID=Europe/Zurich:20230222T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Peter Pickl (Universitä t Tübingen) DESCRIPTION:The derivation of the Vlasov equation from Newtonian mechanics is an old problem in mathematical physics. But while the most interest ing interactions in nature have singularities\, one typically assumes some Lipschitz condition on the interaction force for its microscopic derivation. Recent developments have given results\, where the interac tion force gets singular when the particle number N tends to infinity\, us ually by mollifying or cutting the singularity with a N-dependent moll ifier or cut-off parameter. In the talk I will present most recent develop ments and new results on this topic. X-ALT-DESC:The derivation of the Vlasov equation from Newtonian mechanic
s is an \; \;

old problem in

mathematical physics. But
while the most interesting interactions in \; \;

nature hav
e singularities\,

one typically assumes some Lipschitz condition on
the interaction \; \;

force for its microscopic

deriva
tion. Recent developments have given results\, where the \; \;

interaction force gets singular

when the particle number N tends
to infinity\, usually by mollifying \; \;

or cutting the sin
gularity

with a N-dependent mollifier or cut-off parameter.

In
the talk I will present most recent developments and new results on \
; \;

this topic.

Singular stochastic partial differential equations (SPDEs) of the form \;

\n∂_{t}u \;= \;△u \;+ \
;F \;(u\, ∂_{x}u\, ξ)\, \;

where \;ξ \ ;is an irregular driving noise\, arise in a variety of situations from qua ntum field theory to probability. After introducing some specific examples \, we describe the main difficulty they share\; they are singular due to t he irregularity of the driving noise \;ξ.

\nIn the first part o f the talk we discuss a simple example where using the so-called “Da Pra to-Debusche trick” is sufficient to deal with this difficulty. In the se cond half\, we give a birds-eye view on how regularity structures provide a solution theory for such equations. In particular\, we explain the role of subcriticality (super-renormalisability) and (half) Feynman diagrams in this theory. Lastly\, we shall mention some recent results on the class o f differential operators that are compatible with this general machinery a nd how this relates to the geometry of the underlying space.

DTEND;TZID=Europe/Zurich:20221221T160000 END:VEVENT BEGIN:VEVENT UID:news1419@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20221128T102752 DTSTART;TZID=Europe/Zurich:20221207T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Hyunju Kwon (ETH Zurich) DESCRIPTION:Smooth (spatially periodic) solutions to the incompressible 3D Euler equations have kinetic energy conservation in every local region\, w hile turbulent flows exhibit anomalous dissipation of energy. Toward verif ication of the anomalous dissipation\, Onsager theorem has been establishe d\, which says that the threshold Hölder regularity of the total kinetic energy conservation is 1/3. As a next step\, we discuss a strong Onsager c onjecture\, which combines the Onsager theorem with the local energy inequ ality. X-ALT-DESC:Smooth (spatially periodic) solutions to the incompressible 3 D Euler equations have kinetic energy conservation in every local region\, while turbulent flows exhibit anomalous dissipation of energy. Toward ver ification of the anomalous dissipation\, Onsager theorem has been establis hed\, which says that the threshold Hölder regularity of the total kineti c energy conservation is 1/3. As a next step\, we discuss a strong Onsager conjecture\, which combines the Onsager theorem with the local energy ine quality.

DTEND;TZID=Europe/Zurich:20221207T160000 END:VEVENT BEGIN:VEVENT UID:news1429@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20221117T152426 DTSTART;TZID=Europe/Zurich:20221123T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Michele Dolce (EPFL) DESCRIPTION:Fluids in the ocean are often inhomogeneous\, incompressible an d\, in relevant physical regimes\, can be described by the 2D Euler-Boussi nesq system. Equilibrium states are then commonly observed to be stably st ratified\, namely the density increases with depth. We are interested in c onsidering the case when also a background shear flow is present. In the t alk\, I will describe quantitative results for small perturbations around a stably stratified Couette flow. The density variation and velocity under go an O(1/(t^{1/2})) inviscid damping while the vorticity and density grad ient grow as O(t^{1/2}) in L^2. This is precisely quantified at the linea r level. For the nonlinear problem\, the result holds on the optimal time -scale on which a perturbative regime can be considered. Namely\, given an initial perturbation of size O(eps)\, it is expected that the linear reg ime is observed up to a time-scale O(eps^{-1}). However\, we are able to control the dynamics all the way up to O(eps^{-2})\, where the perturbat ion become of size O(1) due to the linear instability. X-ALT-DESC:Fluids in the ocean are often inhomogeneous\, incompressible and\, in relevant physical regimes\, can be described by the 2D Euler-Bous sinesq system. Equilibrium states are then commonly observed to be stably stratified\, namely the density increases with depth. We are interested in considering the case when also a background shear flow is present. In the talk\, I will describe quantitative results for small perturbations aroun d a stably stratified Couette flow. The density variation and velocity und ergo an O(1/(t^{1/2})) inviscid damping while the vorticity and density gr adient grow as O(t^{1/2}) \;in L^2. This is precisely quantified at th e linear level. \;For the nonlinear problem\, the result holds on the optimal time-scale on which a perturbative regime can be considered. Namel y\, given an initial perturbation of size O(eps)\, \;it is expected th at the linear regime is observed up to a time-scale O(eps^{-1}). However\, \;we are able to control the dynamics all the way up to \;O(eps^{ -2})\, where \;the perturbation become of size O(1) \;due to the l inear instability.

DTEND;TZID=Europe/Zurich:20221123T160000 END:VEVENT BEGIN:VEVENT UID:news1422@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20221107T105340 DTSTART;TZID=Europe/Zurich:20221116T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Nicolas Camps (Universit y of Nantes) DESCRIPTION:Following the seminal work of Bourgain in 1996\, and Burq and T zvetkov in 2008\, a statistical approach to nonlinear dispersive equations has developed in various contexts. We are interested here in Schrödinge r equations with cubic nonlinearity (NLS) in R^d. We first recall the rele vant probabilistic Cauchy theory developed by Bényi\, Oh and Pocovnicu in 2015 in supercritical regimes\, before specifying the norm inflation instability that occurs in this context. The second part is dedicated to long-time dynamics for solutions initiated from these randomized initial d ata. We demonstrate a scattering result that relies on a probabilistic ver sion of the I-method and that allows to solve statistically the scatteri ng conjecture for NLS in dimension 3. Finally\, we present recent develop ments in quasi-linear regimes\, which were initiated by Bringmann in 2019 and which we exploit to exhibit strong solutions to some weakly dispersive equations. This last result is in collaboration with Louise Gassot and Sl im Ibrahim. X-ALT-DESC:Following the seminal work of Bourgain in 1996\, and Burq and
Tzvetkov in 2008\, a statistical approach to nonlinear dispersive equatio
ns has developed in various contexts. \;We are interested here in Schr
ödinger equations with cubic nonlinearity (NLS) in R^d. We first recall t
he relevant probabilistic Cauchy theory \;developed by Bényi\, Oh an
d Pocovnicu in 2015 in supercritical regimes\, before specifying the \
;*norm inflation* \;instability that occurs in this context.&nb
sp\;The second part is dedicated to long-time dynamics for solutions initi
ated from these randomized initial data. We demonstrate a scattering resul
t that relies on a probabilistic version of the \;*I-method*&nb
sp\;and that allows to solve statistically the scattering conjecture for N
LS in dimension 3. \;Finally\, we present recent developments in quasi
-linear regimes\, which were initiated by Bringmann in 2019 and which we e
xploit to exhibit strong solutions to some weakly dispersive equations. Th
is last result is in collaboration with Louise Gassot and Slim Ibrahim.

In the classical theory\, given a vector field $b$ on $\\math bb R^d$\, one usually studies the transport/continuity equation drifted by $b$ looking for solutions in the class of functions (with certain integra bility) or at most in the class of measures. In this seminar I will talk a bout recent efforts\, motivated by the modelling of defects in plastic mat erials\, aimed at extending the previous theory to the case when the unkno wn is instead a family of k-currents in $\\mathbb R^d$\, i.e. generalised $k$-dimensional surfaces. The resulting equation involves the Lie derivati ve $L_b$ of currents in direction $b$ and reads $\\partial_t T_t + L_b T_t = 0$. In the first part of the talk I will briefly introduce this equatio n\, with special attention to its space-time formulation. I will then shif t the focus to some rectifiability questions and Rademacher-type results: given a Lipschitz path of integral currents\, I will discuss the existence of a “geometric derivative”\, namely a vector field advecting the cur rents. Joint work with G. Del Nin and F. Rindler (Warwick).

DTEND;TZID=Europe/Zurich:20221109T160000 END:VEVENT BEGIN:VEVENT UID:news1360@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220625T161018 DTSTART;TZID=Europe/Zurich:20220629T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Luca Fresta (University of Bonn) DESCRIPTION:We consider a system of $N$ interacting fermions initially conf ined in a volume $\\Lambda$. We show that\, in the high-density regime a nd for zero-temperature initial data exhibiting a local semiclassical st ructure\, the solution of the many-body Schrödinger equation can be app roximated by the solution of the nonlinear Hartree equation\, up to erro rs that are small\, for large density\, uniformly in $N$ and $\\Lambda$. This is joint work with M. Porta and B. Schlein. X-ALT-DESC:We consider a system of $N$ interacting fermions initially co
nfined in a \;

volume

$\\Lambda$. We show that\, in the hi
gh-density regime and for zero-temperature \;

initial data

exhibiting a local semiclassical structure\, the solution of the many-bod
y \;

Schrödinger equation can be approximated by the solution o
f the nonlinear \;

Hartree equation\, up to errors that are smal
l\, for large density\, uniformly \;

in $N$ and $\\Lambda$. This
is joint work with M. Porta and B. Schlein.

\;

The Kompaneets equation describes energy transport in low-den
sity (or high \;temperature) plasmas where the dominant energy exchang
e mechanism is \;Compton scattering. The equation itself is a one dime
nsional non-linear \;parabolic equation with a diffusion coefficient t
hat vanishes at the

boundary. This degeneracy\, combined with the no
nlinearity causes an \;out-flux of photons with zero energy\, often in
terpreted as a Bose-Einstein \;condensate. This talk will describe sev
eral results about the long time \;behavior of these equations includi
ng convergence to equilibrium\,

persistence of the condensate\, suff
icient conditions under which it forms\, \;sufficient conditions under
which it doesn't form and a loss formula for \;the mass of the conden
sate.

I will present a recent result concerning global existence fo r the Kuramoto-Sivashinsky equation on the two-dimensional torus with one growing \;mode in each direction. The proof combines PDE techniques wi th a Lyapunov function argument for the growing modes. This is joint work with David Ambrose (Drexel University\, USA).

DTEND;TZID=Europe/Zurich:20220609T160000 END:VEVENT BEGIN:VEVENT UID:news1364@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220606T131917 DTSTART;TZID=Europe/Zurich:20220609T140000 SUMMARY:An afternoon of analysis talks: Giovanni Alberti (University of Pis a) DESCRIPTION:In this talk I will describe some result about the following el ementary problem\, of isoperimetric flavor: Given a set E in R^d with fini te volume\, is it possible to find an hyperplane P that cuts E in two part s with equal volume\, and such that the area of the cut (that is\, the int ersection of P and E) is of the expected order\, namely (vol(E))^{1−1/d} ? We can show that this question\, even in a stronger form\, has a positiv e answer if the dimension d is 3 or higher.But\, interestingly enough\, ou r proof breaks down completely in dimension d=2\, and we do not know the a nswer in this case (but we know that the answer is positive if we allow cu ts that are not exactly planar\, but close to planar). It turns out that this question has some interesting connection with the Kakeya problem. Thi s is a work in progress with Alan Chang (Princeton University). X-ALT-DESC:In this talk I will describe some result about the following
elementary problem\, of isoperimetric flavor:

Given a set E in R^d w
ith finite volume\, is it possible to find an hyperplane P that cuts E in
two parts with equal volume\, and such that the area of the cut (that is\,
the intersection of P and E) is of the expected order\, namely (vol(E))^{
1−1/d}?

We can show that this question\, even in a stronger form\,
has a positive answer if the dimension d is 3 or higher.But\, interesting
ly enough\, our proof breaks down completely in dimension d=2\, and we do
not know the answer in this case (but we know that the answer is positive
if we allow cuts that are not exactly planar\, but close to planar). \
;It turns out that this question has some interesting connection with the
Kakeya problem.

This is a work in progress with Alan Chang (Princeto
n University).

This talk is devoted to the 2D incompressible Euler system in presence of sources and sinks. This model dates back to Viktor Yudovich i n the sixties and is an interesting example of nonlinear open system which has been widely used in controllability theory within the scope of smooth solutions. In this talk we will review how the classical issues of existe nce and uniqueness of weak solutions are challenged by the presence of inc oming and exiting vorticity. \;This talk is based on joint works with Marco Bravin and Florent Noisette. \;

DTEND;TZID=Europe/Zurich:20220525T160000 END:VEVENT BEGIN:VEVENT UID:news1354@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220508T162554 DTSTART;TZID=Europe/Zurich:20220518T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Banhirup Sengupta (Unive rsitat Autònoma de Barcelona) DESCRIPTION:In this talk I am going to provide a pointwise characterisation of nearly incompressible vector fields with bounded curl. Euler vector fi elds fall in this class. I will also talk about rotational properties of E uler flows and nonlinear transport equations involving Cauchy Kernel in th e plane. This is based on joint works with Albert Clop (Barcelona) and Lau ri Hitruhin (Helsinki). X-ALT-DESC:In this talk I am going to provide a pointwise characterisati on of nearly incompressible vector fields with bounded curl. Euler vector fields fall in this class. I will also talk about rotational properties of Euler flows and nonlinear transport equations involving Cauchy Kernel in the plane. This is based on joint works with Albert Clop (Barcelona) and L auri Hitruhin (Helsinki).

DTEND;TZID=Europe/Zurich:20220518T160000 END:VEVENT BEGIN:VEVENT UID:news1327@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220422T115503 DTSTART;TZID=Europe/Zurich:20220504T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Gabriel Sattig (Universi tät Leipzig) DESCRIPTION:It was shown by Modena and Székelyhidi that weak solutions to the incompressible transport equation may be not unique\, even if the tr ansporting field is Sobolev\, thus admitting a unique regular Lagrangian flow. In this talk I will present a recent result saying that non-Lagrang ian solutions are generic in the Baire category sense. Joint work with L . Székelyhidi. X-ALT-DESC:It was shown by Modena and Székelyhidi that weak solutions t o the \;incompressible transport equation may be not unique\, even if the \;transporting field is Sobolev\, thus admitting a unique regular& nbsp\;Lagrangian flow. \;In this talk I will present a recent result s aying that non-Lagrangian \;solutions are generic in the Baire categor y sense. \;Joint work with L. Székelyhidi.

DTEND;TZID=Europe/Zurich:20220504T160000 END:VEVENT BEGIN:VEVENT UID:news1324@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220409T172448 DTSTART;TZID=Europe/Zurich:20220427T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Jaemin Park (University of Barcelona) DESCRIPTION:In this talk\, we study stationary solutions to the 2D incompre ssible Euler equations in the whole plane. It is well-known that any radia l vorticity is stationary. For compactly supported vorticity\, it is more difficult to see whether a stationary solution has to be radial. In the ca se where the vorticity is non-negative\, it has been shown that any statio nary solution has be radial. By allowing the vorticity to change the sign\ , we prove that there exist non-radial stationary patch-type solutions. We construct patch-type solutions whose kinetic energy is infinite or finite . For the finite energy case\, it turns out that a construction of a stati onary solution with compactly supported velocity is possible. X-ALT-DESC:In this talk\, we study stationary solutions to the 2D incomp ressible Euler equations in the whole plane. It is well-known that any rad ial vorticity is stationary. For compactly supported vorticity\, it is mor e difficult to see whether a stationary solution has to be radial. In the case where the vorticity is non-negative\, it has been shown that any stat ionary solution has be radial. By allowing the vorticity to change the sig n\, we prove that there exist non-radial stationary patch-type solutions. We construct patch-type solutions whose kinetic energy is infinite or fini te. For the finite energy case\, it turns out that a construction of a sta tionary solution with compactly supported velocity is possible.

DTEND;TZID=Europe/Zurich:20220427T160000 END:VEVENT BEGIN:VEVENT UID:news1321@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220214T151002 DTSTART;TZID=Europe/Zurich:20220406T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Dr. Eliot Pacherie (NYU Abu Dhabi) DESCRIPTION:tba X-ALT-DESC:tba

DTEND;TZID=Europe/Zurich:20220406T160000 END:VEVENT BEGIN:VEVENT UID:news1313@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220320T140750 DTSTART;TZID=Europe/Zurich:20220323T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Dr. Raphael Winter (ENS Lyon) DESCRIPTION:Following the pioneering work of Lanford\, a rigorous theory ha s been developed for the validation of the Boltzmann equation in the low-d ensity Grad scaling. In the physics literature\, an important issue are th e corrections to the equation for small but positive volume fraction. The first order correction to the Boltzmann equation is conjectured to be give n by the so-called Choh-Uhlenbeck equation\, which is of the form\\r\\n∂ tfϵ=Qϵ\,BE(fϵ\,fϵ)+ϵQϵ\,CU(fϵ\,fϵ\,fϵ).\\r\\n Here Qϵ\,BE is the Boltzmann-Enskog operator\, and the Choh-Uhlenbeck operator Qϵ\,CU is an explicit cubic operator. This operator accounts for the formation of dyna mic microscopic correlations between three particles. In this work\, we pr ove rigorously that the Choh-Uhlenbeck equation gives the first order corr ection to the Boltzmann equation in the Grad-scaling. This is a joint work with Sergio Simonella. X-ALT-DESC:Following the pioneering work of Lanford\, a rigorous theory has been developed for the validation of the Boltzmann equation in the low -density Grad scaling. In the physics literature\, an important issue are the corrections to the equation for small but positive volume fraction. Th e first order correction to the Boltzmann equation is conjectured to be gi ven by the so-called Choh-Uhlenbeck equation\, which is of the form

\n< p>∂tf

Here Qϵ\,BE is the Boltzmann-Enskog operator\, and the C
hoh-Uhlenbeck operator Qϵ\,CU is an explicit cubic operator. This operator
accounts for the formation of dynamic microscopic correlations between th
ree particles. In this work\, we prove rigorously that the Choh-Uhlenbeck
equation gives the first order correction to the Boltzmann equation in the
Grad-scaling. This is a joint work with Sergio Simonella.

\;I will talk about Harris-type theorems and their appli cations to several kinetic equations like the linear BGK\, the linear Bolt zmann\, the kinetic Fokker-Planck equations and some biological kinetic mo dels like the run and tumble equation. Even though the original ideas date back to the 1940s\, the Harris-type arguments recently raised a lot of ma thematical interest \;in the PDE community especially after a simplifi ed proof provided by Hairer and Mattingly in 2011. It is a convenient way to obtain quantifiable convergence rates\, constructive \;proofs and t he existence of a unique \;stationary state comes as a by-product of t he theorems. The latter is especially useful \;for kinetic equations a rising in biology where the shape of the stationary state cannot be known a priori. \;

DTEND;TZID=Europe/Zurich:20220302T160000 END:VEVENT BEGIN:VEVENT UID:news1252@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211115T124620 DTSTART;TZID=Europe/Zurich:20211208T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Martina Zizza (SISSA Tri este) DESCRIPTION:In this talk we tackle the question "How many vector fields are mixing?" analyzing the density properties of divergence-free BV vector fields which are weakly mixing/strongly mixing: this means that their Reg ular Lagrangian Flow is a weakly mixing/strongly mixing measure-preserving map when evaluated at time t=1. More precisely we prove the existence of a G_delta-set U in the space L^1_{t\,x}([0\,1]^3) made of divergence-free vector fields such that:\\r\\n 1) weakly mixing vector fields are a residual G_delta-set in U\;\\r\\n 2) (exponentially fast) strongl y mixing vector fields are a dense subset of U.\\r\\nThe proof of these re sults exploits some connections between ergodic theory and fluid dynamics and it is based on the density of BV vector fields whose Regular Lagrangia n Flow is a permutation of subsquares of the unit square [0\,1]^2 when eva luated at time t=1. X-ALT-DESC:In this talk we tackle the question "How many vector fields a re mixing?" analyzing \; \;the density properties of divergence-fr ee BV vector fields which are weakly mixing/strongly mixing: this means th at their Regular Lagrangian Flow is a weakly mixing/strongly mixing measur e-preserving map when evaluated at time t=1. More precisely we prove the e xistence of a \;G_delta-set U in the space L^1_{t\,x}([0\,1]^3) made o f divergence-free vector fields such that:

\n\; \; 1) \; weakly mixing vector fields are a residual G_delta-set in U\;

\n& nbsp\; \; 2) \; (exponentially fast) strongly mixing vector field s are a dense subset of U.

\nThe proof of these results exploits som e connections between ergodic theory and fluid dynamics and it is based on the density of BV vector fields whose Regular Lagrangian Flow is a permut ation of subsquares of the unit square [0\,1]^2 when evaluated at time t=1 . \;

DTEND;TZID=Europe/Zurich:20211208T160000 END:VEVENT BEGIN:VEVENT UID:news1253@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211122T174531 DTSTART;TZID=Europe/Zurich:20211201T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Elia Bruè (Institute fo r Advanced Study\, Princeton) DESCRIPTION:A long-standing open question in fluid mechanics is whether the Yudovich uniqueness result for the 2d Euler system can be extended to the class of L^p-integrable vorticity. Recently\, there have been formidable attempts to disprove this conjecture\, none of which has by now fully solv ed it. I will outline two possible approaches to this problem. One is bas ed on the convex integration technique introduced by De Lellis and Szekely hidi. The second\, proposed recently by Vishik\, exploits the linear insta bility of certain stationary solutions. X-ALT-DESC:A long-standing open question in fluid mechanics is whether t he Yudovich uniqueness result for the 2d Euler system can be extended to t he class of L^p-integrable vorticity. Recently\, there have been formidabl e attempts to disprove this conjecture\, none of which has by now fully so lved it. \;I will outline two possible approaches to this problem. One is based on the convex integration technique introduced by De Lellis and Szekelyhidi. The second\, proposed recently by Vishik\, exploits the linea r instability of certain stationary \;solutions.

DTEND;TZID=Europe/Zurich:20211201T160000 END:VEVENT BEGIN:VEVENT UID:news1259@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211103T092442 DTSTART;TZID=Europe/Zurich:20211124T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Gioacchino Antonelli (Sc uola Normale Superiore di Pisa) DESCRIPTION:In this talk I will discuss the isoperimetric problem on spaces with curvature bounded from below. I will mainly deal with complete non-c ompact Riemannian manifolds\, but most of the techniques described are met ric in nature and the results could be extended to the case of metric meas ure spaces with synthetic bounds from below on the Ricci tensor\, namely R CD spaces. When the space is compact\, the existence of isoperimetric re gions for every volume is established through a simple application of the direct method of Calculus of Variations. In the noncompact case\, part of the mass could be lost at infinity in the minimization process. Such a mas s can be recovered in isoperimetric regions sitting in limits at infinity of the space. Following this heuristics\, and building on top of results b y Ritoré--Rosales and Nardulli\, I will state a generalized existence res ult for the isoperimetric problem on Riemannian manifolds with Ricci curva ture bounded from below and a uniform bound from below on the volumes of u nit balls. The main novelty in such an approach is the use of the syntheti c theory of curvature bounds to describe in a rather natural way where the mass is lost at infinity. Later\, I will use the latter described general ized existence result to prove new existence criteria for the isoperimetri c problem on manifolds with nonnegative Ricci curvature. In particular\, I will show that on a complete manifold with nonnegative sectional curvatur e and Euclidean volume growth at infinity\, isoperimetric regions exist fo r every sufficiently big volume. Time permitting\, I will describe some fo rthcoming works and some open problems. This talk is based on several pa pers and ongoing collaborations with E. Bruè\, M. Fogagnolo\, S. Nardulli \, E. Pasqualetto\, M. Pozzetta\, and D. Semola. X-ALT-DESC:In this talk I will discuss the isoperimetric problem on spac
es with curvature bounded from below. I will mainly deal with complete non
-compact Riemannian manifolds\, but most of the techniques described are m
etric in nature and the results could be extended to the case of metric me
asure spaces with synthetic bounds from below on the Ricci tensor\, namely
RCD spaces. \;

When the space is compact\, the existence of iso
perimetric regions for every volume is established through a simple applic
ation of the direct method of Calculus of Variations. In the noncompact ca
se\, part of the mass could be lost at infinity in the minimization proces
s. Such a mass can be recovered in isoperimetric regions sitting in limits
at infinity of the space. Following this heuristics\, and building on top
of results by Ritoré--Rosales and Nardulli\, I will state a generalized
existence result for the isoperimetric problem on Riemannian manifolds wit
h Ricci curvature bounded from below and a uniform bound from below on the
volumes of unit balls. The main novelty in such an approach is the use of
the synthetic theory of curvature bounds to describe in a rather natural
way where the mass is lost at infinity. Later\, I will use the latter desc
ribed generalized existence result to prove new existence criteria for the
isoperimetric problem on manifolds with nonnegative Ricci curvature. In p
articular\, I will show that on a complete manifold with nonnegative secti
onal curvature and Euclidean volume growth at infinity\, isoperimetric reg
ions exist for every sufficiently big volume. Time permitting\, I will des
cribe some forthcoming works and some open problems. \;

This tal
k is based on several papers and ongoing collaborations with E. Bruè\, M.
Fogagnolo\, S. Nardulli\, E. Pasqualetto\, M. Pozzetta\, and D. Semola.
DTEND;TZID=Europe/Zurich:20211124T160000
END:VEVENT
BEGIN:VEVENT
UID:news1255@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20211021T155955
DTSTART;TZID=Europe/Zurich:20211103T141500
SUMMARY:Seminar Analysis and Mathematical Physics: In-Jee Jeong (Seoul Nati
onal University)
DESCRIPTION:The evolution of incompressible inviscid fluids is governed by
the Euler equations. We consider the dynamics of vortex rings\, which are
axisymmetric solutions to the three dimensional Euler equations with conce
ntrated axial vorticity. We prove the following infinite norm growth resul
ts: (i) filamentation (formation of a long tail) behavior from a single vo
rtex ring\, and (ii) vortex stretching from the "collision" of two vortex
rings with opposite signs. Joint work with Kyudong Choi (UNIST).
X-ALT-DESC:

The evolution of incompressible inviscid fluids is governed b y the Euler equations. We consider the dynamics of vortex rings\, which ar e axisymmetric solutions to the three dimensional Euler equations with con centrated axial vorticity. We prove the following infinite norm growth res ults: (i) filamentation (formation of a long tail) behavior from a single vortex ring\, and (ii) vortex stretching from the "collision" of two vorte x rings with opposite signs. \;Joint work with Kyudong Choi (UNIST). DTEND;TZID=Europe/Zurich:20211103T160000 END:VEVENT BEGIN:VEVENT UID:news1225@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210908T121913 DTSTART;TZID=Europe/Zurich:20211006T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Emil Wiedemann (Universi tät Ulm) DESCRIPTION:The concept of measure-valued solution was introduced to the th eory of hyperbolic conservation laws by DiPerna in the 1980s. For nonlinea r systems of hyperbolic type\, such as the Euler equations of ideal fluids \, measure-valued solutions are often the only available notion of solutio n\, as the existence of 'honest' solutions is still unknown. Although this relaxation of the solution concept seems like a vast generalization\, whe re a lot of infomation is lost\, it turned out in work of Székelyhidi-W. (2012) that every measure-valued solution can be approximated by weak ones in the incompressible situation. For compressible flows\, however\, the s ituation is much different. I will discuss recent progress in this directi on in joint work with Dennis Gallenmüller. X-ALT-DESC:

The concept of measure-valued solution was introduced to the theory of hyperbolic conservation laws by DiPerna in the 1980s. For nonlin ear systems of hyperbolic type\, such as the Euler equations of ideal flui ds\, measure-valued solutions are often the only available notion of solut ion\, as the existence of 'honest' solutions is still unknown. Although th is relaxation of the solution concept seems like a vast generalization\, w here a lot of infomation is lost\, it turned out in work of Székelyhidi-W . (2012) that every measure-valued solution can be approximated by weak on es in the incompressible situation. For compressible flows\, however\, the situation is much different. I will discuss recent progress in this direc tion in joint work with Dennis Gallenmüller.

DTEND;TZID=Europe/Zurich:20211006T160000 END:VEVENT BEGIN:VEVENT UID:news1190@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210601T082910 DTSTART;TZID=Europe/Zurich:20210602T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Corentin Le Bihan (ENS L yon) DESCRIPTION:A simple model of gas is the hard spheres model. It is a billia rd of little particles which can interact very strongly at very small dist ance (think for example of real billiards with a lot of balls). Because un derstanding such system is an outstanding problem\, people tried to find a limiting process. A first equation governing the density of one particle was given by Boltzmann :∂tf+v⋅∇xf=Q(f\,f)\\r\\nIn its formal derivat ion Boltzmann supposed that two different particles are almost independent \, so the probability of having two particles at the same place is the the product of probability. The validity of such equation is a priori not cle ar since it adds some irreversibly that does not exist in the hard sphere model.\\r\\nLanford solved the problem in its ‘75 paper: Boltzmann’s e quation is true\, up to a time independent of the number of particles (how ever each particle will have in mean less than one collision).\\r\\nNow co mes the question of the boundary. We expect to find some “Lanfords” th eorem even if we add some boundary condition. A first example are the spec ular reflections\, for a deterministic law. An other example\, which would be very important in physics\, is the evolution of a gas between two hot plaques. Then the reflection condition is stochastic. I am interested in a third type of reflection\, also stochastic\, which is a modeling of a rou gh boundary.\\r\\nDuring my talk I will present some ideas of the proof of Boltzmann in the torus R^3/Z^3 and the adaptation in the case of a domain with boundaries. X-ALT-DESC:A simple model of gas is the hard spheres model. It is a bill
iard of little particles which can interact very strongly at very small di
stance (think for example of real billiards with a lot of balls). Because
understanding such system is an outstanding problem\, people tried to find
a limiting process. A first equation governing the density of one particl
e was given by Boltzmann :∂*t**f*+*v*⋅
∇*x**f*=*Q*(*f
*\,*f*)

In its formal derivation Boltzman
n supposed that two different particles are almost independent\, so the pr
obability of having two particles at the same place is the the product of
probability. The validity of such equation is *a priori *not clear
since it adds some irreversibly that does not exist in the hard sphere mod
el.

Lanford solved the problem in its ‘75 paper: Boltzmann’s e quation is true\, up to a time independent of the number of particles (how ever each particle will have in mean less than one collision).

\nNow comes the question of the boundary. We expect to find some “Lanfords” theorem even if we add some boundary condition. A first example are the s pecular reflections\, for a deterministic law. An other example\, which wo uld be very important in physics\, is the evolution of a gas between two h ot plaques. Then the reflection condition is stochastic. I am interested i n a third type of reflection\, also stochastic\, which is a modeling of a rough boundary.

\nDuring my talk I will present some ideas of the pr oof of Boltzmann in the torus R^3/Z^3 and the adaptation in the case of a domain with boundaries.

DTEND;TZID=Europe/Zurich:20210602T160000 END:VEVENT BEGIN:VEVENT UID:news1188@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210510T074110 DTSTART;TZID=Europe/Zurich:20210526T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Bjorn Berntson (KTH Stoc kholm) DESCRIPTION:The half-wave maps (HWM) equation is a recently-introduced inte grable PDE with two distinct relations to the (trigonometric) spin Caloge ro-Moser-Sutherland (CMS) many-body system. Firstly\, the HWM equation ar ises as a certain continuum limit of the CMS system and secondly\, the sol iton solutions of the HWM equation are governed by a complexified version of the CMS system. We present generalizations of the HWM equation that ar e similarly related to the hyperbolic and elliptic spin CMS systems. Th is talk is based on joint work with Rob Klabbers and Edwin Langmann. X-ALT-DESC:The half-wave maps (HWM) equation is a recently-introduced in tegrable PDE with two distinct relations to the (trigonometric) spin \ ;Calogero-Moser-Sutherland (CMS) many-body \;system. Firstly\, the HWM equation arises as a certain continuum limit of the CMS system and second ly\, the soliton solutions of the HWM equation are governed by a complexif ied version of the \;CMS system. We present generalizations of the HWM equation that are similarly related to the \;hyperbolic and elliptic spin \;CMS systems. \;This talk is based on joint work with Rob Kl abbers and Edwin Langmann.

DTEND;TZID=Europe/Zurich:20210526T151500 END:VEVENT BEGIN:VEVENT UID:news1176@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210517T094220 DTSTART;TZID=Europe/Zurich:20210519T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Théotime Girardot (LPMM C) DESCRIPTION:In two-dimensional space there are possibilities for quantum st atistics continuously interpolating between the bosonic and the fermionic one. \\r\\nQuasi-particles obeying such statistics can be described as ord inary bosons and fermions with magnetic interactions. \\r\\nWe study a lim it situation where the statistics/magnetic interaction is seen as a “per turbation from the fermionic end”. \\r\\nWe vindicate a mean-field appro ximation\, proving that the ground state of a gas of anyons is described t o leading order by a semi-classical\, Vlasov-like\, energy functional.\\r\ \nThe ground state of the latter displays anyonic behavior in its momentum distribution. After introducing and stating this result I will give eleme nts of proof based on coherent states\,\\r\\nHusimi functions\, the Diacon is-Freedman theorem and a quantitative version of a semi-classical Pauli p inciple. X-ALT-DESC:In two-dimensional space there are possibilities for quantum statistics continuously interpolating between the bosonic and the fermioni c one.

\nQuasi-particles obeying such statistics can be described a s ordinary bosons and fermions with magnetic interactions.

\nWe stu dy a limit situation where the statistics/magnetic interaction is seen as a “perturbation from the fermionic end”.

\nWe vindicate a mean- field approximation\, proving that the ground state of a gas of anyons is described to leading order by a semi-classical\, Vlasov-like\, energy func tional.

\nThe ground state of the latter displays anyonic behavior i n its momentum distribution. After introducing and stating this result I w ill give elements of proof based on coherent states\,

\nHusimi funct ions\, the Diaconis-Freedman theorem and a quantitative version of a semi- classical Pauli pinciple.

DTEND;TZID=Europe/Zurich:20210519T160000 END:VEVENT BEGIN:VEVENT UID:news1172@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210426T124049 DTSTART;TZID=Europe/Zurich:20210428T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Soeren Petrat (Jacobs Un iversity Bremen) DESCRIPTION:We consider the non-relativistic quantum dynamics of N bosons i n the mean-field scaling limit. It is known that the leading order behavi or is described by the Hartree equation\, and the next-to-leading order b y Bogoliubov theory. Here\, we prove a perturbative expansion around Bog oliubov theory: a norm approximation of the true solution to the Schroedi nger equation to any order in 1/N. The coefficients in the expansion are independent of N\, and can be computed from the solutions to the Hartree and Bogoliubov equations alone. Our expansion leads to approximations of correlation functions and reduced densities to any order in 1/N. In this sense we have completely solved the dynamics of this mean-field model\, a t least for bounded interaction potentials.\\r\\nThis is joint work with L ea Bossmann\, Peter Pickl\, and Avy Soffer. X-ALT-DESC:We consider the non-relativistic quantum dynamics of N bosons in the \;mean-field scaling limit. It is known that the leading order behavior is \;described by the Hartree equation\, and the next-to-lea ding order by \;Bogoliubov theory. Here\, we prove a perturbative expa nsion around \;Bogoliubov theory: a norm approximation of the true sol ution to the \;Schroedinger equation to any order in 1/N. The coeffici ents in the \;expansion are independent of N\, and can be computed fro m the solutions \;to the Hartree and Bogoliubov equations alone. Our e xpansion leads to \;approximations of correlation functions and reduce d densities to any \;order in 1/N. In this sense we have completely so lved the dynamics of \;this mean-field model\, at least for bounded in teraction potentials.

\nThis is joint work with Lea Bossmann\, Peter Pickl\, and Avy Soffer.

DTEND;TZID=Europe/Zurich:20210428T151500 END:VEVENT BEGIN:VEVENT UID:news1162@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210316T094354 DTSTART;TZID=Europe/Zurich:20210407T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Jules Pitcho (University of Zurich) DESCRIPTION:The recent work of Brué\, Colombo and De Lellis has establishe d that\, for Sobolev vector fields\, the continuity equation may be well-p osed in a Lagrangian sense\, yet trajectories of the associated ODE need n ot be unique. We describe how a convex integration scheme for the continui ty equation reveals these degenerate integral curves\; we modify this sche me to produce Sobolev vector fields for which “most” integral curves a re degenerate. More precisely\, we produce Sobolev vector fields which hav e any finite number of integral curves starting almost everywhere. This is a joint work with Massimo Sorella. X-ALT-DESC:The recent work of Brué\, Colombo and De Lellis has establis hed that\, for Sobolev vector fields\, the continuity equation may be well -posed in a Lagrangian sense\, yet trajectories of the associated ODE need not be unique. We describe how a convex integration scheme for the contin uity equation reveals these degenerate integral curves\; we modify this sc heme to produce Sobolev vector fields for which “most” integral curves are degenerate. More precisely\, we produce Sobolev vector fields which h ave any finite number of integral curves starting almost everywhere. This is a joint work with Massimo Sorella. \;

DTEND;TZID=Europe/Zurich:20210407T160000 END:VEVENT BEGIN:VEVENT UID:news1138@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210318T140108 DTSTART;TZID=Europe/Zurich:20210324T140000 SUMMARY:Seminar Analysis and Mathematical Physics: Jonas Lampart (CNRS\, LI CB) DESCRIPTION:I will discuss some properties of the set of all trajectories t hat can be obtained from a fixed initial state by varying the potential in the Schrödinger equation. This is related to the control problem\, i.e. driving the system to a target state\, which turns out to be impossible fo r "typical" target states using bounded potentials. X-ALT-DESC:I will discuss some properties of the set of all trajectories
that can be obtained from a fixed initial state by varying the potential
in the Schrödinger equation.

This is related to the control problem
\, i.e. driving the system to a target state\, which turns out to be impos
sible for "typical" target states using bounded potentials.

The leadi ng-order physics is governed by a Bose-Hubbard Hamiltonian coupling two lo w-energy modes\, each supported in the bottom of one well. Fluctuations be yond these two modes are ruled by two independent Bogoliubov Hamiltonians\ , one for each well.

Our main result is that the variance of the numb er of particles in the low-energy modes is suppressed. This is a violation of the Central Limit Theorem which holds in the occurrence of Bose-Einste in condensation\, and therefore it signals that particles develop correlat ions in the ground state. We achieve our result by proving a precise energ y expansion in term of Bose-Hubbard and Bogoliubov energies.

Joint wo rk with Nicolas Rougerie (ENS Lyon) and Dominique Spehner (Universidad de Concepción). DTEND;TZID=Europe/Zurich:20210317T160000 END:VEVENT BEGIN:VEVENT UID:news1137@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210223T225158 DTSTART;TZID=Europe/Zurich:20210303T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Peter Pickl (LMU Münche n) DESCRIPTION:Abstract: The derivation of effective descriptions from microsc opic dynamics is a very vivid area in mathematical physics. In the talk I will discuss a system of many particles with Newtoni an time evolution that are subject to interaction. It is well k nown that in the weak coupling limit this system converges\, un der smoothness assumption on the interaction force\, to a solut ion of the Vlasov equation. Weakening the types of convergence (con vergence for all initial conditions -> convergence in probabili ty -> convergence in distribution) the smoothness condition on the interaction can be generalized. In the talk I will present recent results in this direction and explain\, which types of c onvergence hold/do not hold under the different assumptions on the interaction force. X-ALT-DESC:

Abstract: The derivation of effective descriptions from micro scopic \; \; \; \; \; \;dynamics is a very vivid a rea in mathematical physics. In the talk I \; \; \; \;&nbs p\; \;will discuss a system of many particles with Newtonian time evol ution \; \; \; \; \; \;that are subject to interac tion. It is well known that in the weak \; \; \; \; \; \;coupling limit this system converges\, under smoothness assumption on \; \; \; \; \; \;the interaction force\, to a s olution of the Vlasov equation. \; \; \; \;Weakening the t ypes of convergence (convergence for all initial \; \; \; \; \; \;conditions ->\; convergence in probability ->\; conver gence in \; \; \; \; \; \;distribution) the smooth ness condition on the interaction can be \; \; \; \; \ ; \;generalized. In the talk I will present recent results in this&nbs p\; \; \; \; \; \;direction and explain\, which types of convergence hold/do not hold \; \; \; \; \; \;u nder the different assumptions on the interaction force.

DTEND;TZID=Europe/Zurich:20210303T161500 END:VEVENT BEGIN:VEVENT UID:news1125@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20201130T113309 DTSTART;TZID=Europe/Zurich:20201216T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Alessandro Goffi (Univer sity of Padova) DESCRIPTION:The problem of maximal regularity in Lebesgue spaces is a heavi ly studied question for linear PDEs that dates back to Calderón and Zygmu nd for the Poisson equation and the potential theoretic approach developed by Ladyzhenskaya et al for the heat equation\, and represents the corners tone in the analysis of many nonlinear PDEs. After a brief overview of the linear theory\, in this talk I will focus on maximal L^q-regularity for v iscous Hamilton-Jacobi equations with superlinear first order terms. I wil l first survey on recent results obtained for the stationary equation\, wh ich answer positively to a conjecture raised by P.-L. Lions\, via a Bernst ein-type argument. Then\, I will discuss Lipschitz and optimal L^q-regular ity for parabolic Hamilton-Jacobi equations that instead are tackled throu gh a refinement of the Evans’ nonlinear adjoint method\, thus exploiting fine regularity properties for advection-diffusion equations with “roug h” drifts\, providing new regularity results for systems of PDEs arising in the theory of Mean Field Games. These are joint works with Marco Ciran t (Padova). X-ALT-DESC:The problem of maximal regularity in Lebesgue spaces is a hea vily studied question for linear PDEs that dates back to Calderón and Zyg mund for the Poisson equation and the potential theoretic approach develop ed by Ladyzhenskaya et al for the heat equation\, and represents the corne rstone in the analysis of many nonlinear PDEs. After a brief overview of t he linear theory\, in this talk I will focus on maximal L^q-regularity for viscous Hamilton-Jacobi equations with superlinear first order terms. I w ill first survey on recent results obtained for the stationary equation\, which answer positively to a conjecture raised by P.-L. Lions\, via a Bern stein-type argument. Then\, I will discuss Lipschitz and optimal L^q-regul arity for parabolic Hamilton-Jacobi equations that instead are tackled thr ough a refinement of the Evans’ nonlinear adjoint method\, thus exploiti ng fine regularity properties for advection-diffusion equations with “ro ugh” drifts\, providing new regularity results for systems of PDEs arisi ng in the theory of Mean Field Games. These are joint works with Marco Cir ant (Padova). \;

DTEND;TZID=Europe/Zurich:20201216T160000 END:VEVENT BEGIN:VEVENT UID:news1126@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20201122T211830 DTSTART;TZID=Europe/Zurich:20201209T150000 SUMMARY:Seminar Analysis and Mathematical Physics: Mickaël Latocca (ENS Pa ris) DESCRIPTION:The possible growth of the Sobolev norms of the solution to the 2d (and 3d) Euler equations and its quantification remains ill-known. Th e only general bound is double exponential. Conversely\, such a double e xponential growth scenario occurs for specific initial data in the settin g of the disc (Kiselev-Sverak). In the setting of the torus\, only an exp onentially growing scenario has been exhibited (Zlatos). Could the double exponential scenario occur on the torus? What is the typical behaviour t hat could be expected? It is highly possible that on the torus\, Sobolev norms generically do not grow fast.In this talk\, I will present some resu lts obtained in this direction. We will construct invariant measures for the 2d Euler equation at high regularity ($H^s$\, $s>2$) and prove that o n the support of the measure\, Sobolev norms do not grow faster than poly nomially.Refining the method allows to construct an invariant measure to t he 3d Euler equations at high regularity ($H^s$\, $s>7/2$) and thus const ructglobal dynamics on the support of the measure\, exhibiting at most po lynomial growth.Finally\, it time permits we will discuss the properties o f the measures constructed. X-ALT-DESC:The possible growth of the Sobolev norms of the solution to the& nbsp\;2d (and 3d) Euler equations and its quantification remains ill-known . The \;only general bound is double exponential. Conversely\, such a double \;exponential growth scenario occurs for specific initial data in the \;setting of the disc (Kiselev-Sverak). In the setting of the t orus\, only an \;exponentially growing scenario has been exhibited (Zl atos). Could the \;double exponential scenario occur on the torus? Wha t is the typical \;behaviour that could be expected? It is highly poss ible that on the torus\, \;Sobolev norms generically do not grow fast.In this talk\, I will present some results obtained in this di rection. We \;will construct invariant measures for the 2d Euler equat ion at high \;regularity ($H^s$\, $s>\;2$) and prove that on the sup port of the measure\, \;Sobolev norms do not grow faster than polynomi ally.

Refining the method allows to construct an invariant meas ure to the 3d \;Euler equations at high regularity ($H^s$\, $s>\;7/2 $) and thus construct

global dynamics on the support of the measure\, exhibiting at most \;polynomial growth.

Finally\, it time permits we will discuss the properties of the measures \;constructed. DTEND;TZID=Europe/Zurich:20201209T160000 END:VEVENT BEGIN:VEVENT UID:news1106@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20201116T100656 DTSTART;TZID=Europe/Zurich:20201209T140000 SUMMARY:Seminar Analysis and Mathematical Physics: Lea Bossmann (IST Austri a) DESCRIPTION:Abstract. \\r\\nWe consider a system of N bosons in the mean-f ield scaling regime in an external trapping potential. We derive an asymp totic expansion of the low-energy eigenstates and the corresponding ener gies\, which provides corrections to Bogoliubov theory to any order in 1/ N. We show that the structure of the ground state and of the non-degenera te low-energy eigenstates is preserved by the dynamics ifthe external trap is switched off. This talk is based on joint works with Sören Petrat\, Peter Pickl\, Robert Seiringer\, and Avy Soffer (arXiv:1912.11004 and arX iv:2006.09825). X-ALT-DESC:

Abstract. \;

\nWe consider a system of N bosons in
the mean-field scaling \;regime in an external trapping potential. We
derive an asymptotic \;expansion of the low-energy eigenstates and the
corresponding \;energies\, which provides corrections to Bogoliubov t
heory to any order \;in 1/N. We show that the structure of the ground
state and of the \;non-degenerate low-energy eigenstates is preserved
by the dynamics if

the external trap is switched off. This talk is ba
sed on joint works \;with Sören Petrat\, Peter Pickl\, Robert Seiring
er\, and Avy Soffer \;(arXiv:1912.11004 and arXiv:2006.09825).

Abstract: \; \; \;

We consider N spin 1/2 f
ermions interacting with a positive \;

and regular enough potenti
al in three dimensions. We compute the ground \;

state energy of
the system in the dilute regime at second order in the \;

particl
e density. We recover a well-know expression for the ground state \;**energy which depends on the interaction potentials only via its \;
scattering length. A first proof of this result has been given by Li
eb\, \;Seiringer and Solovej. We discuss a new derivation of thi
s formula \;which \; makes use of the almost-bosonic nature
of the low-energy \;excitations of the systems. Based on a joint
work with Marco Falconi\, \;Christian Hainzl\, Marcello Porta.<
/p>
DTEND;TZID=Europe/Zurich:20201125T161500
END:VEVENT
BEGIN:VEVENT
UID:news1100@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20201102T110041
DTSTART;TZID=Europe/Zurich:20201111T141500
SUMMARY:Seminar Analysis and Mathematical Physics: Lars Eric Hientzsch (Ins
titut Fourier\, University of Grenoble Alpes)
DESCRIPTION:The quantum Navier-Stokes (QNS) equations describe a compressib
le fluid including a degenerate density dependent viscosity and a dispersi
ve tensor accounting for capillarity effects. The system can be seen as vi
scous correction of the Quantum Hydrodynamics (QHD) arising e.g. as protot
ype model in the description of superfluidity. We consider the (QNS) syste
m on the whole space with non-trivial farfield behaviour providing the sui
table framework to study coherent structures and the incompressible limit.
\\r\\nFirst\, we prove global existence of finite energy weak solutions (F
EWS) in dimension two and three. To compensate for the lack of control of
the velocity field around vacuum regions\, we construct approximate soluti
ons to a truncated formulation of (QNS) on a sequence of invading domains.
Suitable compactness properties are inferred from the Bresch-Desjadins en
tropy estimates. This is joint work with P. Antonelli and S. Spirito.\\r\\
nSecond\, we address the low Mach number limit for FEWS to the (QNS) syste
m (in collaboration with P. Antonelli and P. Marcati). The main novelty is
a precise analysis of the acoustic dispersion altered by the presence of
the dispersive capillarity tensor. The linearised system is governed by
the Bogoliubov dispersion relation. The desired decay of the acoustic part
follows from refined Strichartz estimates.
X-ALT-DESC:**

**The quantum Navier-Stokes (QNS) equations describe a compress
ible fluid including a degenerate density dependent viscosity and a disper
sive tensor accounting for capillarity effects. The system can be seen as
viscous correction of the Quantum Hydrodynamics (QHD) arising e.g. as prot
otype model in the description of superfluidity. We consider the (QNS) sys
tem on the whole space with non-trivial farfield behaviour providing the s
uitable framework to study coherent structures and the incompressible limi
t.**

First\, we prove global existence of finite energy weak solutio ns (FEWS) in dimension two and three. To compensate for the lack of contro l of the velocity field around vacuum regions\, we construct approximate s olutions to a truncated formulation of (QNS) on a sequence of invading dom ains. Suitable compactness properties are inferred from the Bresch-Desjadi ns entropy estimates. This is joint work with P. Antonelli and S. Spirito.

\nSecond\, we address the low Mach number limit for FEWS to the (QN S) system (in collaboration with P. Antonelli and P. Marcati). The main no velty is a precise analysis of the acoustic dispersion altered by the pres ence of the dispersive capillarity tensor. The \;linearised \;syst em is governed by the Bogoliubov dispersion relation. The desired decay of the acoustic part follows from refined Strichartz estimates. \;

DTEND;TZID=Europe/Zurich:20201111T160000 END:VEVENT BEGIN:VEVENT UID:news1036@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20201024T154014 DTSTART;TZID=Europe/Zurich:20201104T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Marco Falconi (Universit y of Roma Tre) DESCRIPTION:Abstract. \\r\\nIn this talk I will overview variational proble ms arising from the study ofquantum matter interacting with a macroscopic force field. These interactions are verycommon in both solid state and con densed matter physics\, as well as in higher energysettings. In particular \, I will focus on the link between the effective and microscopicdescripti on of such variational problems\, using techniques of quasi-classical anal ysisdeveloped in recent years in collaboration with M. Correggi and M. Oli vieri. X-ALT-DESC:Abstract.

\nIn this talk I will overview variational p roblems arising from the study ofquantum matter interacting with a macrosc opic force field. These interactions are verycommon in both solid state an d condensed matter physics\, as well as in higher energysettings. In parti cular\, I will focus on the link between the effective and microscopicdesc ription of such variational problems\, using techniques of quasi-classical analysisdeveloped in recent years in collaboration with M. Correggi and M . Olivieri.

DTEND;TZID=Europe/Zurich:20201104T160000 END:VEVENT BEGIN:VEVENT UID:news1086@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20201007T090901 DTSTART;TZID=Europe/Zurich:20201021T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Maria Teresa Chiri (Penn State University) DESCRIPTION:A posteriori Error Estimates for Numerical Solutions to Hyperbo lic Conservation Laws X-ALT-DESC:A posteriori Error Estimates for Numerical Solutions to Hyperbol ic Conservation Laws DTEND;TZID=Europe/Zurich:20201021T160000 END:VEVENT BEGIN:VEVENT UID:news1087@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20200930T171430 DTSTART;TZID=Europe/Zurich:20201014T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Silja Haffter (EPFL) DESCRIPTION:The surface quasigeostrophic equation (SGQ) is a 2d physical mo del equation which emerges in meteorology. It has attracted the attention of the mathematical community since it shares many of the essential diffic ulties of 3d fluid dynamics: in the supercritical regime for instance\, wh ere dissipation is modelled by a fractional Laplacian of order less than 1 /2\, it is not known whether or not smooth solutions blow-up in finite tim e. On the other hand\, the scheme of Leray still produces global-in-time w eak solutions from any L^2-initial datum\, but their regularity is poorly understood. In this talk\, I will propose a nonempty notion of "suitable w eak solution" for the supercritical SQG equation and prove that those solu tions are smooth outside a compact set of quantifiable Hausdorff dimension \; in particular they are smooth almost everywhere. I will also give a con jecture on what we believe to be an optimal dimension estimate. This is a joint work with Maria Colombo (EPFL). X-ALT-DESC:The surface quasigeostrophic equation (SGQ) is a 2d physical model equation which emerges in meteorology. It has attracted the attentio n of the mathematical community since it shares many of the essential diff iculties of 3d fluid dynamics: in the supercritical regime for instance\, where dissipation is modelled by a fractional Laplacian of order less than 1/2\, it is not known whether or not smooth solutions blow-up in finite t ime. On the other hand\, the scheme of Leray still produces global-in-time weak solutions from any L^2-initial datum\, but their regularity is poorl y understood. In this talk\, I will propose a nonempty notion of "\;su itable weak solution"\; for the supercritical SQG equation and prove t hat those solutions are smooth outside a compact set of quantifiable Hausd orff dimension\; in particular they are smooth almost everywhere. I will a lso give a conjecture on what we believe to be an optimal dimension estima te. \;This is a joint work with Maria Colombo (EPFL).

DTEND;TZID=Europe/Zurich:20201014T160000 END:VEVENT BEGIN:VEVENT UID:news1088@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20200923T091416 DTSTART;TZID=Europe/Zurich:20201007T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Klaus Widmayer (EPFL) DESCRIPTION:A point charge is a particularly basic and important equilibriu m of the Vlasov-Poisson equations\, and the study of its stability has ins pired several major contributions. In this talk we present some recent wor k\, which brings a fresh perspective on this problem. Our new approach com bines a Lagrangian analysis of the linearized problem with an Eulerian PDE framework in the nonlinear analysis\, all the while respecting the symple ctic structure. As a result\, for the case of radial initial data\, we see that solutions are global and in fact disperse to infinity via a modified scattering along trajectories of the linearized flow. This is joint work with Benoit Pausader (Brown University). X-ALT-DESC:A point charge is a particularly basic and important equilibr ium of the Vlasov-Poisson equations\, and the study of its stability has i nspired several major contributions. In this talk we present some recent w ork\, which brings a fresh perspective on this problem. Our new approach c ombines a Lagrangian analysis of the linearized problem with an Eulerian P DE framework in the nonlinear analysis\, all the while respecting the symp lectic structure. As a result\, for the case of radial initial data\, we s ee that solutions are global and in fact disperse to infinity via a modifi ed scattering along trajectories of the linearized flow. This is joint wor k with Benoit Pausader (Brown University).

DTEND;TZID=Europe/Zurich:20201007T160000 END:VEVENT BEGIN:VEVENT UID:news1002@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20200910T202733 DTSTART;TZID=Europe/Zurich:20200930T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Simone Dovetta (CNR-IMAT I\, Pavia) DESCRIPTION:The talk overviews some recent developments about nonlinear Sch roedinger equations (NLS) on metric graphs. Precisely\, we concentrate on variational problems for the NLS energy functional subject to the mass con straint. After a brief recap of the well-known behaviour of such model on the real line\, we address the existence of NLS ground states on noncompac t metric graphs\, with a specific focus on periodic graphs and infinite tr ees. The emergence of threshold phenomena rooted in the nature of these gr aphs is discussed. Finally\, we provide some insights on the uniqueness of ground states at fixed mass. On the one hand\, uniqueness is shown to hol d for two classes of graphs with halflines. On the other hand\, a countere xample to uniqueness in full generality is exhibited. The matter we discus s is part of a wider research line\, developed in collaboration with sever al authors. The results explicitly covered by the talk refer to a series o f papers\, some of which are joint works with Riccardo Adami\, Enrico Serr a and Paolo Tilli. X-ALT-DESC:The talk overviews some recent developments about nonlinear S chroedinger equations (NLS) on metric graphs. Precisely\, we concentrate o n variational problems for the NLS energy functional subject to the mass c onstraint. After a brief recap of the well-known behaviour of such model o n the real line\, we address the existence of NLS ground states on noncomp act metric graphs\, with a specific focus on periodic graphs and infinite trees. The emergence of threshold phenomena rooted in the nature of these graphs is discussed. Finally\, we provide some insights on the uniqueness of ground states at fixed mass. On the one hand\, uniqueness is shown to h old for two classes of graphs with halflines. On the other hand\, a counte rexample to uniqueness in full generality is exhibited. The matter we disc uss is part of a wider research line\, developed in collaboration with sev eral authors. The results explicitly covered by the talk refer to a series of papers\, some of which are joint works with Riccardo Adami\, Enrico Se rra and Paolo Tilli.

DTEND;TZID=Europe/Zurich:20200930T160000 END:VEVENT BEGIN:VEVENT UID:news1022@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20200217T114353 DTSTART;TZID=Europe/Zurich:20200226T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Luigi De Rosa (EPF Lausa nne) DESCRIPTION:I will describe how convex integration techniques can be used t o show that wilddissipative solutions of the incompressible Euler equation s are typical in the Baire category sense. This also partially solves a co njecture by Philip Isett on the sharpness of the kinetic energy regularity for Hölder continuous solutions of the Euler equations. The talk will b e based on a recent work obtained in collaboration with Riccardo Tione. X-ALT-DESC:I will describe how convex integration techniques can be used to show that wilddissipative solutions of the incompressible Euler equati ons are typical in the Baire category sense. This also partially solves a conjecture by Philip Isett on the sharpness of the kinetic energy regulari ty \;for Hölder continuous solutions of the Euler equations. The talk will be based on a recent work obtained \;in collaboration with Ricca rdo Tione.

DTEND;TZID=Europe/Zurich:20200226T160000 END:VEVENT BEGIN:VEVENT UID:news929@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20191015T153025 DTSTART;TZID=Europe/Zurich:20191023T141500 SUMMARY:Seminar Analysis and Mathematical Physics: Fabian Ziltener (Utrecht University) DESCRIPTION:The goal of this talk is to show in an example how analysis and symplectic geometry are related in several ways.Symplectic geometry origi nated from classical mechanics\, where the canonical symplectic form on ph ase space appears in Hamilton's equation. A (smooth) diffeomorphism on a s ymplectic manifold is called a symplectomorphism iff it preserves the symp lectic form. This happens iff the diffeomorphism solves a certain inhomoge neous quadratic first order system of PDE's. In classical mechanics symple ctomorphisms play the role of canonical transformations.A famous result by Eliashberg and Gromov states that the set of symplectomorphisms is $C^0$- closed in the set of all diffeomorphisms. This is remarkable\, since in ge neral\, the $C^0$-limit of a sequence of solutions of a first order system of PDE's need not solve the system. A well-known proof of the Eliashberg- Gromov theorem is based on Gromov's symplectic nonsqueezing theorem for ba lls.In my talk I will sketch this proof. Furthermore\, I will present a sy mplectic nonsqueezing result for spheres that sharpens Gromov's theorem. T he proof of this result is based on the existence of a holomorphic map fro m the (real) two-dimensional unit disk to a certain symplectic manifold\, satisfying some Lagrangian boundary condition. Such a map solves the Cauch y-Riemann equation for a certain almost complex structure. X-ALT-DESC:The goal of this talk is to show in an example how analysis and symplectic geometry are related in several ways.Symplectic geo metry originated from classical mechanics\, where the canonical symplectic form on phase space appears in Hamilton's equation. A (smooth) diffeomorp hism on a symplectic manifold is called a symplectomorphism iff it preserv es the symplectic form. This happens iff the diffeomorphism solves a certa in inhomogeneous quadratic first order system of PDE's. In classical mecha nics symplectomorphisms play the role of canonical transformations.

< br />A famous result by Eliashberg and Gromov states that the set of sympl ectomorphisms is $C^0$-closed in the set of all diffeomorphisms. This is r emarkable\, since in general\, the $C^0$-limit of a sequence of solutions of a first order system of PDE's need not solve the system. A well-known p roof of the Eliashberg-Gromov theorem is based on Gromov's symplectic nons queezing theorem for balls.

In my talk I will sketch this proof . Furthermore\, I will present a symplectic nonsqueezing result for sphere s that sharpens Gromov's theorem. The proof of this result is based on the existence of a holomorphic map from the (real) two-dimensional unit disk to a certain symplectic manifold\, satisfying some Lagrangian boundary con dition. Such a map solves the Cauchy-Riemann equation for a certain almost complex structure. DTEND;TZID=Europe/Zurich:20191023T160000 END:VEVENT BEGIN:VEVENT UID:news833@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190529T183455 DTSTART;TZID=Europe/Zurich:20190529T141500 SUMMARY:Seminar Analysis: Mikaela Iacobelli (ETH Zürich) DESCRIPTION:The Vlasov-Poisson system is a kinetic equation that models col lisionless plasma. A plasma has a characteristic scale called the Debye le ngth\, which is typically much shorter than the scale of observation. In t his case the plasma is called ‘quasineutral’. This motivates studying the limit in which the ratio between the Debye length and the observation scale tends to zero. Under this scaling\, the formal limit of the Vlasov-P oisson system is the Kinetic Isothermal Euler system.The Vlasov-Poisson sy stem itself can formally be derived as the limit of a system of ODEs descr ibing the dynamics of a system of N interacting particles\, as the number of particles approaches infinity. The rigorous justification of this mean field limit remains a fundamental open problem.In this talk we present how the mean field and quasineutral limits can be combined to derive the Kine tic Isothermal Euler system from a regularised particle model. X-ALT-DESC:\nThe Vlasov-Poisson system is a kinetic equation that models co llisionless plasma. A plasma has a characteristic scale called the Debye l ength\, which is typically much shorter than the scale of observation. In this case the plasma is called ‘quasineutral’. This motivates studying the limit in which the ratio between the Debye length and the observation scale tends to zero. Under this scaling\, the formal limit of the Vlasov- Poisson system is the Kinetic Isothermal Euler system.

The Vlasov-Poi sson system itself can formally be derived as the limit of a system of ODE s describing the dynamics of a system of N interacting particles\, as the number of particles approaches infinity. The rigorous justification of thi s mean field limit remains a fundamental open problem.

In this talk w e present how the mean field and quasineutral limits can be combined to de rive the Kinetic Isothermal Euler system from a regularised particle model . DTEND;TZID=Europe/Zurich:20190529T160000 END:VEVENT BEGIN:VEVENT UID:news888@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190515T102013 DTSTART;TZID=Europe/Zurich:20190522T141500 SUMMARY:Analysis Seminar: Armin Schikorra (University of Pittsburgh) DESCRIPTION:The degree of a map between two spheres of same dimension can b e estimated by Sobolev norm of said map (of the right class). In this ta lk I will discuss to what extend this is possible for the Hopf degree as well – and why the estimate we have is “analytically optimal” but p robably not “topologically optimal”. Joint work with J. Van Schafting en. X-ALT-DESC:\nThe degree of a map between two spheres of same dimension can be \;estimated by Sobolev norm of said map (of the right \;class). In this talk I will discuss to what extend this is possible \;for the Hopf degree as well – and why the estimate we \;have is “analytic ally optimal” but probably not “topologically \;optimal”. Joint work with J. Van Schaftingen. DTEND;TZID=Europe/Zurich:20190522T160000 END:VEVENT BEGIN:VEVENT UID:news856@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190410T101051 DTSTART;TZID=Europe/Zurich:20190424T141500 SUMMARY:Seminar Analysis: Giorgio Stefani (Scuola Normale Superiore di Pisa ) DESCRIPTION:Somewhat surprisingly\, the first appearance of the concept of a fractional derivative is found in a letter written to de l'Hôpital by L eibniz in 1695. Since then\, Fractional Calculus has fascinated generation s of mathematicians and several definitions of fractional derivatives have appeared. In more recent years\, the fractional operator defined as the g radient of the Riesz potential has received particular attention\, since i t has revealed to be a useful tool for the study of fractional-order PDEs and fractional Sobolev spaces. In a joint work with G. E. Comi\, combining the PDE approach developed by Spector and his collaborators with the dist ributional point of view adopted by Šilhavý\, we introduced new notions of fractional variation and fractional Caccioppoli perimeter in analogy w ith the classical BV theory. Within this framework\, we were able to parti ally extend De Giorgi’s Blow-up Theorem to sets of locally finite fracti onal Caccioppoli perimeter\, proving existence of blow-ups and giving a fi rst characterisation of these (possibly non-unique) limit sets. In this ta lk\, after a quick overview on Fractional Calculus\, I will introduce the main features of the fractional operators involved and then give an accoun t on the main results on the fractional variation we were able to achieve so far. X-ALT-DESC:\nSomewhat surprisingly\, the first appearance of the concept of a fractional derivative is found in a letter written to de l'Hôpital by Leibniz in 1695. Since then\, Fractional Calculus has fascinated generatio ns of mathematicians and several definitions of fractional derivatives hav e appeared. In more recent years\, the fractional operator defined as the gradient of the Riesz potential has received particular attention\, since it has revealed to be a useful tool for the study of fractional-order PDEs and fractional Sobolev spaces. In a joint work with G. E. Comi\, combinin g the PDE approach developed by Spector and his collaborators with the dis tributional point of view adopted by Šilhavý\, \;we introduced new n otions of fractional variation and fractional Caccioppoli perimeter in ana logy with the classical BV theory. Within this framework\, we were able to partially extend De Giorgi’s Blow-up Theorem to sets of locally finite fractional Caccioppoli perimeter\, proving existence of blow-ups and givin g a first characterisation of these (possibly non-unique) limit sets. In t his talk\, after a quick overview on Fractional Calculus\, I will introduc e the main features of the fractional operators involved and then give an account on the main results on the fractional variation we were able to ac hieve so far. DTEND;TZID=Europe/Zurich:20190424T160000 END:VEVENT BEGIN:VEVENT UID:news828@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190327T120936 DTSTART;TZID=Europe/Zurich:20190410T141500 SUMMARY:Seminar Analysis: Dominik Inauen (University of Zurich) DESCRIPTION:The problem of embedding abstract Riemannian manifolds isometri cally (i.e. preserving the lengths) into Euclidean space stems from the co nceptually fundamental question of whether abstract Riemannian manifolds a nd submanifolds of Euclidean space are the same. As it turns out\, such em beddings have a drastically different behaviour at low regularity (i.e. C^ 1) than at high regularity (i.e. C^2): for example\, it's possible to find C^1 isometric embeddings of the standard 2-sphere into arbitrarily small balls in R^3\, and yet\, in the C^2 category there is (up to translation a nd rotation) just one isometric embedding\, namely the standard inclusion. Analoguous to the Onsager conjecture\, one might ask if there is a regula rity threshold in the Hölder scale which distinguishes these behaviours. In my talk I will give an overview of what is known concerning the latter question. X-ALT-DESC:\nThe problem of embedding abstract Riemannian manifolds isometr ically (i.e. preserving the lengths) into Euclidean space stems from the c onceptually fundamental question of whether abstract Riemannian manifolds and submanifolds of Euclidean space are the same. As it turns out\, such e mbeddings have a drastically different behaviour at low regularity (i.e. C ^1) than at high regularity (i.e. C^2): for example\, it's possible to fin d C^1 isometric embeddings of the standard 2-sphere into arbitrarily small balls in R^3\, and yet\, in the C^2 category there is (up to translation and rotation) just one isometric embedding\, namely the standard inclusion . Analoguous to the Onsager conjecture\, one might ask if there is a regul arity threshold in the Hölder scale which distinguishes these behaviours. In my talk I will give an overview of what is known concerning the latter question. \; DTEND;TZID=Europe/Zurich:20190410T160000 END:VEVENT BEGIN:VEVENT UID:news830@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190327T081509 DTSTART;TZID=Europe/Zurich:20190327T141500 SUMMARY:Seminar Analysis: Tobias Weth (University of Frankfurt) DESCRIPTION:I will report on some recent results - obtained in joint work w ith Huyuan Chen - on Dirichlet problems for the Logarithmic Laplacian Oper ator\, which arises as formal derivative of fractional Laplacians at order s= 0. I will discuss the functional analytic framework for these problems and show how it allows to characterize the asymptotics of principal Diric hlet eigenvalues and eigenfunctions of fractional Laplacians as the order tends to zero. Furthermore\, I will discuss necessary and sufficient condi tions on domains giving rise to weak and strong maximum principles for the logarithmic Laplacian. If time permits\, I will also discuss regularity e stimates for solutions to corresponding Poisson problems. X-ALT-DESC:\nI will report on some recent results - obtained in joint work with Huyuan Chen - on Dirichlet problems for the Logarithmic Laplacian Ope rator\, which arises as formal derivative of fractional Laplacians at orde r s= 0. I will discuss the functional analytic framework for these problem s and show how it allows to characterize the asymptotics of principal Diri chlet eigenvalues and eigenfunctions of fractional Laplacians as the order tends to zero. Furthermore\, I will discuss necessary and sufficient cond itions on domains giving rise to weak and strong maximum principles for th e logarithmic Laplacian. If time permits\, I will also discuss regularity estimates for solutions to corresponding Poisson problems. DTEND;TZID=Europe/Zurich:20190327T160000 END:VEVENT BEGIN:VEVENT UID:news329@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181113T174428 DTSTART;TZID=Europe/Zurich:20181128T141500 SUMMARY:Seminar Analysis: Stefano Spirito (Università dell'Aquila) DESCRIPTION:In this talk I will present some results concerning the analysi s of finite energy weak solutions of the Navier-Stokes-Korteweg equations\ , which model the dynamic of a viscous compressible fluid with diffuse int erface. A general theory of global existence is still missing\, however for some particular cases of physical interest I will present results rega rding the global existence and the compactness of finite energy weak solut ions. The talk is based on a series of joint works with Paolo Antonelli (G SSI - Gran Sasso Science Institute\, L’Aquila). X-ALT-DESC:\nIn this talk I will present some results concerning the analys is of finite energy weak solutions of the Navier-Stokes-Korteweg equations \, which model the dynamic of a viscous compressible fluid with diffuse in terface. \;A general theory of global existence is still missing\, ho wever for some particular cases of physical interest I will present result s regarding the global existence and the compactness of finite energy weak solutions. The talk is based on a series of joint works with Paolo Antone lli (GSSI - Gran Sasso Science Institute\, L’Aquila). DTEND;TZID=Europe/Zurich:20181128T160000 END:VEVENT BEGIN:VEVENT UID:news95@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181113T174437 DTSTART;TZID=Europe/Zurich:20181121T141500 SUMMARY:Seminar Analysis: Elia Bruè (Scuola Normale Superiore di Pisa) DESCRIPTION:Since the work by DiPerna and Lions (1989) the continuity and t ransport equation under mild regularity assumptions on the vector field ha ve been extensively studied\, becoming a florid research field. The applic ability of this theory is very wide\, especially in the study of partial d ifferential equations and very recently also in the field of non-smooth ge ometry.\\r\\nThe aim of this talk is to give an overview of the quantitati ve side of the theory initiated by Crippa and De Lellis. We address the pr oblem of mixing and propagation of regularity for solutions to the continu ity equation drifted by Sobolev fields. The problem is well understood whe n the vector field enjoys a Sobolev regularity with integrability exponent p>1 and basically nothing is known (at the quantitative level) in the cas e p=1.\\r\\nWe present sharp regularity estimates for the case p>1 and new attempts to attack the challenging question in the case p=1. This is a jo in work with Quoc-Hung Nguyen. X-ALT-DESC:\nSince the work by DiPerna and Lions (1989) the continuity and transport equation under mild regularity assumptions on the vector field h ave been extensively studied\, becoming a florid research field. The appli cability of this theory is very wide\, especially in the study of partial differential equations and very recently also in the field of non-smooth g eometry.\nThe aim of this talk is to give an overview of the quantitative side of the theory initiated by Crippa and De Lellis. We address the probl em of mixing and propagation of regularity for solutions to the continuity equation drifted by Sobolev fields. The problem is well understood when t he vector field enjoys a Sobolev regularity with integrability exponent p& gt\;1 and basically nothing is known (at the quantitative level) in the ca se p=1.\nWe present sharp regularity estimates for the case p>\;1 and ne w attempts to attack the challenging question in the case p=1. This is a j oin work with Quoc-Hung Nguyen. DTEND;TZID=Europe/Zurich:20181121T160000 END:VEVENT BEGIN:VEVENT UID:news423@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181217T184454 DTSTART;TZID=Europe/Zurich:20171204T160000 SUMMARY:Seminar Analysis: Xavier Ros-Oton (University of Zürich) DESCRIPTION:We present a brief overview of the regularity theory f or free boundaries in different obstacle problems. We describe ho w a monotonicity formula of Almgren plays a central role in the study of the regularity of the free boundary in some of these problems. Finally \, we explain new strategies which we have recently developed to deal wit h cases in which monotonicity formulas are not available. X-ALT-DESC:\nWe present a brief overview of the regularity theory for free boundaries in different obstacle problems. We describe h ow a monotonicity formula of Almgren plays a central role in the study of the regularity of the free boundary in some of these problems. Finall y\, we explain new strategies which we have recently developed to deal wi th cases in which monotonicity formulas are not available. DTEND;TZID=Europe/Zurich:20171204T180000 END:VEVENT BEGIN:VEVENT UID:news422@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181211T232725 DTSTART;TZID=Europe/Zurich:20171127T160000 SUMMARY:Seminar Analysis: Jérémy Sok (University of Basel) DESCRIPTION:We consider Dirac operators on the 3-sphere with singular magne tic fields which are supported on links\, that is on one-dimensional manif olds which are diffeomorphic to finitely many copies of S1. Each connected component carries a flux 2πα which exhibits a 2π-periodicity\, just li ke Aharonov-Bohmsolenoids in the complex plane. We study the kernel of su ch operators through the spectral flow of loops corresponding to tuning so me flux from 0 to 2π\, that is the number of eigenvalues crossing 0 along the loop (counted algebraically). It turns out that the spectral flow is generically non-zero and depends on the shape of the curves and their link ing number. Through the stereographic projection the result extends to R3. And then by smearing out the magnetic fields we obtain new solutions (ψ\ ,A) to the zero-mode equation on R3:\\r\\nσ·(-i∇+A)=0\,(ψ\,A) ∈ H1( R3)2 × \\dot{H}1(R3)3 ∩ L6(R3)3\,\\r\\nwhere σ=(σ)j=1...3 denotes the family of the Pauli matrices\, A is the magnetic potential associated to the magnetic field ∇×A\, and σ⋅(-i∇+A) is the corresponding Dirac operator in R3.\\r\\n(Joint work with Fabian Portmann and Jan Philip Solov ej) X-ALT-DESC:\nWe consider Dirac operators on the 3-sphere with singular magn etic fields which are supported on links\, that is on one-dimensional mani folds which are diffeomorphic to finitely many copies of S

(ψ\,A) ∈ H

\;& nbsp\; u = 0 \; \; \; \; \; \; \; \; \ ; \; \; on ∂B\nwhere B is the unit ball of R

\;&nbs p\; \; \; \; \; \; \; \; \; \; \;& nbsp\; \; \; \; \; \; φ(x)=x \; \; \; &nb sp\; \; x∈∂Ω.\nLocal case. We first consider the (local) existenc e\, uniqueness and optimal regularity for the problem\ng