Ground state (in-)stability and long-time behavior in multi-dimensional Schrödinger equations]]>

Cusp formation of singular vortex patches

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Rigidity of critical points of degenerate polyconvex energies

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

The third law of black hole thermodynamics and the black hole formation threshold

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Zero-dispersion limit for the Benjamin-Ono Equation]]>

Sharp nonuniqueness of the transport equation with Sobolev vector fields

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Volumetric approach to intermittency in fully developed turbulence

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Newton-Nash iterations in PDE

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Dynamics of mean-field Fermi systems with nonzero pairing]]>

Model hierarchies in semiclassical semi-relativistic quantum physics: From Dirac-Maxwell to Euler-Poisson]]>

semiclassical semi-relativistic quantum dynamics. It is at the

center of a hierarchy of models from the Dirac-Maxwell equation to

the Euler-Poisson equation that are linked by asymptotic analysis of

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

We discuss the models and their application in plasma and

accelerator physics as well as the many mathematical problems they

pose.

]]>Landau damping near the Poisson equilibrium in R^3

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Permanent features of 2d fluids

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**Notice the special time!**

Steady vortex rings with surface tension

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Boltzmann equation, hard sphere systems and their small and large deviations]]>

Boltzmann equation, hard sphere systems and their small and large deviations]]>

Microscopic derivation of a nonlinear Schrödinger equation with a nonlinear point interaction in 1D]]>

We assume introducing a tiny impurity located at the origin and considering that the interaction between every pair of bosons is mediated by the impurity through a three-body interaction. Moreover, by assuming short-range scaling and choosing an initial fully condensed state, we prove convergence of one-particle density operators in the trace-class topology. This is the first derivation of the so-called nonlinear delta model. This research is a collaborative work with Prof. Riccardo Adami. ]]>

Linear Inviscid Damping in the 2D Euler-Boussinesq system in the periodic channel

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

A mean field planning approach to regularizations of the optimal transport problem

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Assuming the initial and terminal measures to have densities, we prove that the optimal curve remains positive and locally bounded in time. We focus on the case that the transport problem is set on a compact Riemannian manifold with Ricci curvature bounded below.

The approach follows ideas introduced by P.L. Lions in the theory of mean-field games about optimization problems with penalizing congestion terms. Crucial steps of our strategy include displacement convexity properties in the Eulerian approach and the analysis of distributional subsolutions to Hamilton-Jacobi equations.

The result provides a smooth approximation of Wasserstein-2 geodesics.

]]>The intertwined derivative Schrödinger system of Calogero-Moser type. (Online Seminar via Zoom).]]>

Scale invariant bounds for mixing in the Rayleigh-Taylor instability

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

The free energy of the dilute Bose gas]]>

While at temperature close to zero we have precise information 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 critical point. In this talk I will discuss how thermal excitations can be described by Bogoliubov theory, allowing us to estimate the free energy of the Bose gas in the Gross-Pitaevskii regime.

This is joint work with A. Deuchert and D. Stocker.

]]>Instantaneous loss of regularity for SQG with fractional diffusion

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Examples of non-uniqueness for the advection-diffusion equation

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This is part of my Master's thesis, supervised by Massimo Sorella and Maria Colombo. ]]>

The strong Onsager conjecture

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Traveling waves near shear flows

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On the Kelvin-Voigt model for Viscoelastic material

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Regularity of the Pressure in the Incompressible Euler Equation in a Bounded Domain

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The Dirac equation in dispersive PDEs: advances and open problems]]>

]]>

Invariant KAM tori for the planar Euler equations

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Derivation of the Vlasov equation for singular interactions]]>

old problem in

mathematical physics. But while the most interesting 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

interaction force gets singular

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

or cutting the singularity

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 PDEs

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

∂_{t}u = △u + F (u, ∂_{x}u, ξ),

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

In the first part of the talk we discuss a simple example where using the so-called “Da Prato-Debusche trick” is sufficient to deal with this difficulty. In the second 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 of differential operators that are compatible with this general machinery and how this relates to the geometry of the underlying space.

]]>A strong Onsager conjecture on the Euler equations

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Long-time behaviour in the 2D Euler-Boussinesq equations near a stably stratified Couette flow

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Some probabilistic approaches for NLS in the Euclidean space

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Transport of currents and geometric Rademacher-type theorems

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Effective Dynamics of Extended Fermi Gases at High Density]]>

volume

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

initial data

exhibiting a local semiclassical structure, the solution of the many-body

Schrödinger equation can be approximated by the solution of the nonlinear

Hartree equation, up to errors that are small, for large density, uniformly

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

]]>

Bose-Einstein condensation in the Kompaneets equation

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

boundary. This degeneracy, combined with the nonlinearity causes an out-flux of photons with zero energy, often interpreted as a Bose-Einstein condensate. This talk will describe several results about the long time behavior of these equations including convergence to equilibrium,

persistence of the condensate, sufficient conditions under which it forms, sufficient conditions under which it doesn't form and a loss formula for the mass of the condensate. ]]>

Global existence for the 2D Kuramoto-Sivashinsky equation with growing modes

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Dividing a set in half

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Given a set E in R^d with 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, interestingly 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 (Princeton University). ]]>

2D incompressible Euler system in presence of sources and sinks

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Pointwise description of Eulerian vector fields and rotational properties of their flows

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Genericity of non-Lagrangian solutions to the transport equation

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Existence of non-radial stationary solutions to the 2D Euler equation

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

]]>

Rigorous validity of the Choh-Uhlenbeck equation ]]>

∂tfϵ=Qϵ,BE(fϵ,fϵ)+ϵQϵ,CU(fϵ,fϵ,fϵ).

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 dynamic microscopic correlations between three 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.

Obtaining quantitative hypocoercivity estimates by using Harris-type theorems.]]>

Mixing properties of BV vector fields

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

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

2) (exponentially fast) strongly mixing vector fields are a dense subset of U.

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

]]>Nonuniqueness results for the 2d Euler equations

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The isoperimetric problem on spaces with curvature bounded from below

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When the space is compact, the existence of isoperimetric regions 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 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 with 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 described generalized existence result to prove new existence criteria for the isoperimetric problem on manifolds with nonnegative Ricci curvature. In particular, I will show that on a complete manifold with nonnegative sectional curvature and Euclidean volume growth at infinity, isoperimetric regions exist for every sufficiently big volume. Time permitting, I will describe some forthcoming works and some open problems.

This talk is based on several papers and ongoing collaborations with E. Bruè, M. Fogagnolo, S. Nardulli, E. Pasqualetto, M. Pozzetta, and D. Semola. ]]>

Instabilities in vortex ring dynamics

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Measure-Valued Solutions for the Compressible Euler Equations

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Boltzmann-Grad limit of a hard sphere system in a box with diffusive boundary conditions]]>

In its formal derivation 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 clear since it adds some irreversibly that does not exist in the hard sphere model.

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

Now 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 specular 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 rough boundary.

During 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.

]]>Generalizations of the half-wave maps equation]]>

Semiclassical limit for almost fermionic anyons]]>

Quasi-particles obeying such statistics can be described as ordinary bosons and fermions with magnetic interactions.

We study a limit situation where the statistics/magnetic interaction is seen as a “perturbation from the fermionic end”.

We 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 functional.

The ground state of the latter displays anyonic behavior in its momentum distribution. After introducing and stating this result I will give elements of proof based on coherent states,

Husimi functions, the Diaconis-Freedman theorem and a quantitative version of a semi-classical Pauli pinciple.

]]>All Orders of the Dynamics of Bosons in the Mean-field Limit ]]>

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

]]>Almost everywhere non-uniqueness for integral curves of Sobolev vector fields

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

The trajectories of 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 for "typical" target states using bounded potentials. ]]>

Bosons in a double well: two-mode approximation and fluctuations]]>

The leading-order physics is governed by a Bose-Hubbard Hamiltonian coupling two low-energy modes, each supported in the bottom of one well. Fluctuations beyond these two modes are ruled by two independent Bogoliubov Hamiltonians, one for each well.

Our main result is that the variance of the number 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-Einstein condensation, and therefore it signals that particles develop correlations in the ground state. We achieve our result by proving a precise energy expansion in term of Bose-Hubbard and Bogoliubov energies.

Joint work with Nicolas Rougerie (ENS Lyon) and Dominique Spehner (Universidad de Concepción). ]]>

Derivation of the Vlasov equation: Different types of convergence]]>

Optimal regularity for viscous Hamilton-Jacobi equations in Lebesgue spaces

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

High-regularity Invariant Measures for 2d and 3d Euler Equations and Growth of the Sobolev Norms

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

In this talk, I will present some results obtained in this direction. We will construct invariant measures for the 2d Euler equation at high regularity ($H^s$, $s>2$) and prove that on the support of the measure, Sobolev norms do not grow faster than polynomially.

Refining the method allows to construct an invariant measure 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. ]]>

Asymptotic expansion of low-energy excitations for weakly interacting bosons

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

We 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 theory 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 based on joint works with Sören Petrat, Peter Pickl, Robert Seiringer, and Avy Soffer (arXiv:1912.11004 and arXiv:2006.09825).

Radial symmetry in stationary/uniformly-rotating solutions to 2D Euler equation

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

The dilute Fermi gas via Bogoliubov theory

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We consider N spin 1/2 fermions interacting with a positive

and regular enough potential in three dimensions. We compute the ground

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

particle 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 Lieb,

Seiringer and Solovej. We discuss a new derivation of this 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. ]]>

Existence of weak solutions and incompressible limit for quantum Navier-Stokes equations

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

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

Second, we address the low Mach number limit for FEWS to the (QNS) system (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.

]]>Variational Problems in Quasi-Classical Systems

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

In this talk I will overview variational problems arising from the study ofquantum matter interacting with a macroscopic force field. These interactions are verycommon in both solid state and condensed matter physics, as well as in higher energysettings. In particular, I will focus on the link between the effective and microscopicdescription of such variational problems, using techniques of quasi-classical analysisdeveloped in recent years in collaboration with M. Correggi and M. Olivieri.

]]>A posteriori Error Estimates for Numerical Solutions to Hyperbolic Conservation Laws

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Are weak solutions to the supercritical surface quasigeostrophic equation smooth a.e.?

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

On the stability of a point charge for the Vlasov-Poisson system

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Variational problems for nonlinear Schroedinger equations on metric graphs

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Sharp energy regularity and typical wild solutions of the incompressible Euler equations

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

C^0-closedness of the set of symplectomorphisms, spherical symplectic nonsqueezing, and holomorphic maps

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Symplectic geometry originated from classical mechanics, where the canonical symplectic form on phase space appears in Hamilton's equation. A (smooth) diffeomorphism on a symplectic manifold is called a symplectomorphism iff it preserves the symplectic form. This happens iff the diffeomorphism solves a certain inhomogeneous quadratic first order system of PDE's. In classical mechanics symplectomorphisms 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 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 proof of the Eliashberg-Gromov theorem is based on Gromov's symplectic nonsqueezing theorem for balls.

In my talk I will sketch this proof. Furthermore, I will present a symplectic nonsqueezing result for spheres 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 condition. Such a map solves the Cauchy-Riemann equation for a certain almost complex structure. ]]>

Recent results on singular limits for Vlasov-Poisson

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The Vlasov-Poisson system is a kinetic equation that models collisionless plasma. A plasma has a characteristic scale called the Debye length, 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-Poisson system itself can formally be derived as the limit of a system of ODEs describing 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 Kinetic Isothermal Euler system from a regularised particle model.

Thresholds for Measuring Degree in fractional Sobolev Spaces

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The 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 “analytically optimal” but probably not “topologically optimal”. Joint work with J. Van Schaftingen.

]]>A distributional approach to fractional Sobolev spaces and fractional variation

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Somewhat 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 generations of mathematicians and several definitions of fractional derivatives have 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, combining the PDE approach developed by Spector and his collaborators with the distributional point of view adopted by Šilhavý, we introduced new notions of fractional variation and fractional Caccioppoli perimeter in analogy 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 giving a first characterisation of these (possibly non-unique) limit sets. In this talk, after a quick overview on Fractional Calculus, I will introduce 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 achieve so far.

]]>Flexibility and Rigidity of Isometric Embeddings

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

The problem of embedding abstract Riemannian manifolds isometrically (i.e. preserving the lengths) into Euclidean space stems from the conceptually fundamental question of whether abstract Riemannian manifolds and submanifolds of Euclidean space are the same. As it turns out, such embeddings 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 and rotation) just one isometric embedding, namely the standard inclusion. Analoguous to the Onsager conjecture, one might ask if there is a regularity 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.

]]>The Dirichlet Problem for the Logarithmic Laplacian

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

I will report on some recent results - obtained in joint work with Huyuan Chen - on Dirichlet problems for the Logarithmic Laplacian Operator, 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 Dirichlet eigenvalues and eigenfunctions of fractional Laplacians as the order tends to zero. Furthermore, I will discuss necessary and sufficient conditions on domains giving rise to weak and strong maximum principles for the logarithmic Laplacian. If time permits, I will also discuss regularity estimates for solutions to corresponding Poisson problems.

]]>Finite Energy Weak Solutions of the Navier-Stokes-Korteweg equations

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

In this talk I will present some results concerning the analysis of finite energy weak solutions of the Navier-Stokes-Korteweg equations, which model the dynamic of a viscous compressible fluid with diffuse interface. A general theory of global existence is still missing, however for some particular cases of physical interest I will present results regarding the global existence and the compactness of finite energy weak solutions. The talk is based on a series of joint works with Paolo Antonelli (GSSI - Gran Sasso Science Institute, L’Aquila).

]]>Propagation of regularity and mixing estimates for solutions of the incompressible continuity equation with Sobolev vector field

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Since the work by DiPerna and Lions (1989) the continuity and transport equation under mild regularity assumptions on the vector field have been extensively studied, becoming a florid research field. The applicability of this theory is very wide, especially in the study of partial differential equations and very recently also in the field of non-smooth geometry.

The 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 problem of mixing and propagation of regularity for solutions to the continuity equation drifted by Sobolev fields. The problem is well understood when the vector field enjoys a Sobolev regularity with integrability exponent p>1 and basically nothing is known (at the quantitative level) in the case p=1.

We 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 join work with Quoc-Hung Nguyen.

]]>Regularity of free boundaries in obstacle problems for integro-differential operators

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We present a brief overview of the regularity theory for free boundaries in different obstacle problems. We describe how 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 with cases in which monotonicity formulas are not available.

]]>Dirac operators with magnetic links

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We consider Dirac operators on the 3-sphere with singular magnetic fields which are supported on links, that is on one-dimensional manifolds which are diffeomorphic to finitely many copies of S^{1}. Each connected component carries a flux 2πα which exhibits a 2π-periodicity, just like Aharonov-Bohmsolenoids in the complex plane. We study the kernel of such operators through the spectral flow of loops corresponding to tuning some 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 linking number. Through the stereographic projection the result extends to R^{3}. And then by smearing out the magnetic fields we obtain new solutions (ψ,A) to the zero-mode equation on R^{3}:

σ·(-i∇+A)=0,

(ψ,A) ∈ H^{1}(R^{3})^{2} × \dot{H}^{1}(R^{3})^{3 }∩ L^{6}(R^{3})^{3},

where 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 R^{3}.

(Joint work with Fabian Portmann and Jan Philip Solovej)

]]>External boundary control of the motion of a rigid body immersed in a perfect two-dimensional fluid

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We consider the motion of a rigid body due to the pressure of a surrounded two-dimensional irrotational perfect incompressible fluid, the whole system being confined in abounded domain with an impermeable condition on a part of the external boundary. Thanks to an impulsive control strategy we prove that there exists an appropriate boundary condition on the remaining part of the external boundary (allowing some fluid going in and out the domain) such that the immersed rigid body is driven from some given initial position and velocity to some final position and velocity in a given positive time, without touching the external boundary. The controlled part of the external boundary is assumed to have a nonvoid interior and the final position is assumed to be in the same connected component of the set of possible positions as the initial position.

]]>Patterns formation for minimizers of a local/non-local interaction functional in general dimension

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We study a functional consisting of a perimeter term and a non-local term which are in competition, both in the discrete and continuous setting.In the discrete setting such functional was introduced by Giuliani, Lebowitz, Liebe and Seiringer. For both the continuous and discrete problem, we show that the global minimizers are exact periodic stripes. One striking feature of the functionals is that the minimizers are invariant under a smaller group of symmetries than the functional itself. In the continuous setting, to our knowledge this is the first example of a model with local/nonlocal terms in competition such that the functional is invariant under permutation of coordinates and the minimizers display a pattern formation which is one dimensional. Such behaviour for a smaller range of exponents in the discrete setting had been already shown,using different techniques. This is a joint work with E. Runa.

]]>The Porous Medium Equation with Large Initial Data on Negatively Curved Riemannian Manifolds

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We discuss existence and uniqueness of very weak solutions of the Cauchy problem for the porous medium equation on Cartan–Hadamard manifolds satisfying suitable lower bounds on Ricci curvature, with initial data that can grow at inﬁnity at a prescribed rate, that depends crucially on the curvature bounds. Furthermore, we give a precise estimate for the maximal existence time, and we show that in general solutions do not exist if the initial data grow at inﬁnity too fast. Such results have been recently obtained jointly with G. Grillo and M. Muratori.

]]>Finite Energy Weak Solutions of the Quantum Navier-Stokes equations

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

In this talk I will present a result concerning the global existence of finite energy weak solutions of the quantum Navier-Stokes equations. The novelty of the result is that we are able to consider the vacuum in the definition of weak solutions. The main tools are a new formulation of the equations which allows us to get an additional a priori estimate to prove compactness and a non trivial choice of the approximation system consistent with the a priori estimates.

This is a joint work with Paolo Antonelli (GSSI)

]]>Nonlocal obstacle problems: regularity of the solutions and the free boundaries

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

We will introduce some recent results in collaboration with L. Caffarelli and X. Ros-Otonon the optimal regularity of the solutions and the regularity of the free boundaries (near regular points) for nonlocal obstacle problems.The main novelty is that we obtain results for different operators than the fractional Laplacian. Indeed, we can consider infinitesimal generators of non rotationally invariant stable L ́evy processes.

]]>Blowup for Fractional NLS

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

In this talk, we give sufficient criteria for blowup of solutions to nonlocal Schrödinger equations with focusing power-type nonlinearity. To give an outline of the arguments used in the proof, let us mainly focus on the mass-supercritical problem posed on the whole space R^{n} with prescribed radial initial datum of negative energy.

This is a joint work with Thomas Boulenger and Enno Lenzmann

]]>Rademacher’s Theorem, Cheeger’s Conjecture and PDEs for measures

Tags: TAG Events Forschung Informatik, TAG Events DMI]]>

The classical Rademacher Theorem asserts that every Lipschitz function is differentiablea lmost everywhere with respect to Lebesgue measure. On the other hand, Preiss (’90) gave a surprising example of a nullset in the plane such that every Lipschitz function is differentiable at at least one point of this set. Thus, it is a natural question to ask whether there exists a singular measure such that all Lipschitz functions are differentiable with respect to this singular measure. It turns out that this question has an intricate connection to the geometric structure of normal one-currents. In this talk I will present a converse to Rademacher’s Theorem, which settles the question in the negative in all dimensions: if a positive measure μ has the property that all Lipschitz functions are μ-a.e. differentiable, then μ is absolutely continuous with respect to Lebesgue measure (in the plane, this question was already solved by Alberti, Csornyei and Preiss in ’05). In a geometric context, Cheeger conjectured in ’99 that in all Lipschitz differentiability spaces (which are essentially Lipschitz manifolds in which Rademacher’s Theorem holds) likewise there is a “functional converse” to Rademacher’s Theorem. As the second main result, I will present a recent solution to this conjecture.Technically, the proofs of both of these theorems are based on a recent structure result for the singular parts of PDE-constrained measures, its corollary on the structure of normalone-currents, and the powerful theory of Alberti representations.

This is a joint work with A. Marchese and G. De Philippis

]]>Fractional div-curl quantities and applications to nonlocal geometric equations

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We investigate a fractional notion of gradient and divergence operator. We generalize the div-curl estimate by Coifman-Lions-Meyer-Semmes to fractional div-curl quantities. We demonstrate how these quantities appear naturally in nonlocal geometric equations, which can be used to obtain a regularity theory for fractional harmonic maps and critical systems with nonlocal antisymmetric potential.

This is a joint work with Armin Schikorra

]]>Partial Regularity for Holonomic Minimisers of Quasiconvex Functionals

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We prove partial regularity for local minimisers of certain strictly quasiconvex integral functionals, over a class of Sobolev mappings into a compact Riemannian manifold, to which such mappings are said to be holonomically constrained. Several applications to variational problems in condensed matter physics with broken symmetries will also be discussed related to the manifold constraint condition.

]]>A Morse index formula for the Lane-Emden problem

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We consider the semilinear Lane-Emden problem (E_{p}):

-Δu = |u|^{p-1}u in B

u = 0 on ∂B

where B is the unit ball of R^{N}, N≥3, centered at the origin and 1< p < p_{S}, p_{S}=(N+2)/(N−2). We compute the Morse index of any radial solutionup of (E_{p}), for p sufficiently close to p_{S}. The proof exploits the asymptotic behavior of u_{p} as p→p_{S} and the analysis of a limit eigenvalue problem.

This is a joint work with F. De Marchis and F. Pacella

]]>Variational aspects of Liouville equations and systems on compact surfaces

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A class of Liouville equations and systems on compact surfaces is considered: we focus on the Toda system which is motivated in mathematical physics by the study of models in non-abelian Chern-Simons theory and in geometry in the description of holomorphic curves in complex analysis. We discuss its variational aspects which yield existence results.

]]>From condensed matter physics to probability theory

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The basic laws governing atoms and electrons are well understood, but it is impossible to make predictions about the behaviour of large systems of condensed matter physics. A popular approach is to introduce simple models and to use notions of statistical mechanics. I will review quantum spin systems and their stochastic representations in terms of random permutations and random loops. I will also describe the ‘universal’ behaviour that is common to loop models in dimensions 3 and more.

]]>Groundstate solutions for a nonlinear Choquard equation

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I will discuss the existence of groundstate solutions for the Choquard equation in the whole space R^{N}. I will first consider the case of a homogeneous nonlinearity F(u) =|u|^{p}, then I will prove the existence of solutions under general hypotheses. In particular, the cases N=2 and N≥3 will have to be treated differently. The solutions are found through a variational mountain pass strategy.

Uniqueness of weak solutions to transport equation with two-dimensional nearly incompressible BV vector field

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Given a bounded, autonomous vector field b: R^{2} → R^{2}, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation

(1) ∂tu + b · ∇u = 0.

We prove that uniqueness of weak solutions holds under the assumptions that b is of class BV and it is nearly incompressible. Our proof is based on a splitting technique (introduced previously by Alberti, Bianchini and Crippa) that allows to reduce (1) to a family of 1-dimensional equations which can be solved explicitly, thus yielding uniqueness for the original problem.

In order to perform this program, we use Disintegration Theorem and known results on the structure of level sets of Lipschitz maps: this is done after a suitable localization of the problem, in which we exploit also Ambrosio’s superposition principle.

This is joint work with S. Bianchini and N. A. Gusev.

]]>BMO, quasiconformal maps, and transport equations

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We will explain two results about (linear and nonlinear) transport equations, quasiconformal maps, and vector fields with unbounded divergence. Originally, these results are motivated by a difficult problem on Muckenhoupt weights and elliptic PDE. However, classical harmonic analysis tools allow to reformulate this problem in variational BMO terms, and then a theorem by H. M. Reimann brings naturally the connection to the transport theory.

]]>Stability and Regularity Properties of the Navier-Stokes Equation

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We will discuss some regularity properties of weak solutions to the three-dimensional Navier–Stokes equation. We will first recall the classical partial regularity theory, developed by Scheffer and later by Caffarelli–Kohn–Nirenberg. Then we will present some new results in both the small data and perturbative frameworks.

]]>Eulerian and Lagrangian continuous solutions to a non-convex balance law

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We consider continuous solutions to the single balance law ∂_{t}u+∂_{x}(f(u)) = g, g bounded, f ∈ C^{2}. We discuss correspondences among the source terms in the Eulerian and Lagrangian settings, extending previous works relative to the flux f(u) = u^{2} when possible. Counterexamples point out a new behavior of solutions when f is non-convex, and when the set of inflection points of f is not negligible, stressing the difference among the Lagrangian/Eulerian formulations in this context.

This is a joint work with G. Alberti and S. Bianchini.

]]>Scalar conservation laws with discontinuous flux

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In order to obtain uniqueness for solutions of scalar conservation laws with discontinuous flux, Kruzhkov’s entropy conditions are not enough and additional dissipation conditions have to be imposed on the discontinuity set of the flux. Understanding these conditions requires to study the structure of solutions on the discontinuity set. I will show that under quite general assumptions on the flux, solutions admit traces on the discontinuity set of the flux. This allows to show that any pair of solutions satises a Kato type inequality with an explicit reminder term concentrated on the discontinuities of the flux. Applications to uniqueness is then discussed.

]]>On the concentration of entropy dissipation for scalar conservation laws

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After a brief overview about the classical well-posedness result for scalar conservation laws, we investigate the structure of bounded solutions. In particular we prove that the entropy dissipation measure is concentrated on a countably 1-rectifiable set. In order to prove this result we introduce the notion of Lagrangian representation of the solution.

This is a joint work with Stefano Bianchini.

]]>On linear inviscid damping, boundary effects and blow-up

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The Euler equations of fluid dynamics are time-reversible equations and possess many conserved quantities, including the kinetic energy and entropy. Furthermore, as shown by Arnold, they even have the structure of an infinite-dimensional Hamiltonian system. Despite these facts, in experiments one observes a damping phenomenon for small velocity perturbations to monotone shear flows, where the perturbations decay with algebraic rates. In this talk, I discuss the underlying phase-mixing mechanism of linear inviscid damping, its mathematical challenges and how to establish decay with optimal rates for a general class of monotone shear flows. Here, a particular focus will be on the setting of a channel with impermeable walls, where boundary effects asymptotically result in the formation of singularities.

]]>Möbius invariant versions of the Willmore flow and related evolution equations

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While the Willmore energy is invariant under Möbius transformations, its negative L^{2}-gradient flow is not - simply because the L^{2}-scalar product used in its definition does not have this invariance. In this talk we present Möbius invariant versions of the Willmore flow picking up ideas of Ruben Jakob and Oded Schramm. We will discuss its uses and limitations and prove well-posedness of the Cauchy problem and attractivity of local minimizers.

Networks of curves in the plane moving by curvature

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I consider the motion by curvature of a network of curves in the Euclidean plane and I discuss existence, uniqueness, and asymptotic behavior of the evolution. In particular, I focus on two model cases: a regular embedded network composed by three curves with fixed endpoints (triod) and a regular embedded network composed by two curves, one of which is closed (spoon). After talking about the state of art of the problem, I will present some new and possibly ”incoming” results obtained with Carlo Mantegazza and Matteo Novaga.

]]>On Moser type inequalities in the whole space

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The Trudinger-Moser inequality is a substitute for the well known Sobolev embedding theorem when the limiting case is considered. We discuss Moser type inequalities in the whole space which involve complete and reduced Sobolev norm.Then we investigate the optimal growth rate of the exponential type function both in the first order case and in the higher order case.

]]>The finiteness problem for minimal surfaces of bounded index in a 3-manifold

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Given a closed, Riemannian 3-manifold (N, g) without symmetries (more precisely: generic) and a non-negative integer p, can we say something about the number of minimal surfaces it contains whose Morse index is bounded by p? More realistically, can we prove that such number is necessarily finite? This is the classical ”generic finiteness” problem, which has a rich history and exhibits interesting subtleties even in its basic counterpart concerning closed geodesics on surfaces. We settle such question when g is a bumpy metric of positive scalar curvature by proving that either finiteness holds or N does contain a copy of RP^{3} in its prime decomposition and we discuss the obstructions to any further generalisation of such result. When g is assumed to be strongly bumpy (meaning that all closed, immersed minimal surfaces do not have Jacobi fields, a notion recently proved to be generic by White) then the finiteness conclusion is true for any compact 3-manifold without boundary.

Nonlinear Bounds in Hölder Spaces for the Monge-Ampère Equation

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It is well known that for many second-order PDEs the solution v gains two derivatives with respect to the right-hand side g in Hölder spaces. Often, however, it is useful to have a quantitative understanding of regularity. In ’89, Caffarelli proved interior a priori estimates for fully nonlinear, uniformly elliptic equations. Specifically, he showed that ‖v‖_{C2,α(B_{1/2})}≤C(‖v‖_{L∞(B_{1})}+‖g‖_{Cα(B_{1})}) and C∼1/α as α→0. The natural question to ask is then: Can one extend such quantitative estimates to other equations? An equation that appears frequently in analysis, geometry, and applications is the Monge-Ampére equation det(D^{2}u) = f. The Monge-Ampère equation enjoys the same qualitative regularity gains as its linear counterpart the Poisson equation in the appropriate setting, and so we ask whether or not the quantitative picture is also the same. This is not the case. In this talk, we will first review Caffarelli’s interior a priori estimates. Then, we will move to the Monge-Ampère equation and see a different picture.

(Joint work with Alessio Figalli and Connor Mooney)

]]>Regularity issues for some nonlocal and nonlinear elliptic equations

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In this talk, I will review some regularity results for weak solutions of nonlocal variants of the p-Laplace equation. The model case is given by the Euler-Lagrange equation of an Aronszajn–Gagliardo–Slobodeckij seminorm. In particular, I will present a higher differentiability result for solutions, recently obtained in collaboration with Erik Lindgren (KTH). I will also discuss some connections of these equations to an Optimal Transport problem with congestion effects.

]]>Lipschitz extensions of maps between metric spaces

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A pair of metric spaces (X;Y) is said to have the Lipschitz extension property if any Lipschitz map from a subset of X into Y can be extended to a globally defined Lipschitz map to the whole space X. In this talk I will first recall some classical extension results for spaces with a linear structure, and I will present recent results for the case when the target space Y is the Heisenberg group.

]]>Local eigenvalue statistics for random regular graphs

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I discuss results on local eigenvalue statistics for random regular graphs. Under mild growth assumptions on the degree, we prove that the local semicircle law holds at the optimal scale, and that the bulk eigenvalue statistics coincide with those of the GOE from random matrix theory.

(Joint work with R. Bauerschmidt, J. Huang and H.-T. Yau.)

]]>Invariant manifolds for the porous medium equation

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We investigate the speed of convergence and higher-order asymptotics of solutions to the porous medium equation. Applying a nonlinear change of variables, we rewrite the equation as a diffusion on a fixed domain with quadratic nonlinearity.The degeneracy is cured by viewing the dynamics on a hypocycloidic manifold. It is in this framework that we can prove a differentiable dependency of solutions on the initial data, and thus, dynamical systems methods are applicable. Our main result is the construction of invariant manifolds in the phase space of solutions which are tangent at the origin to the eigenspaces of the linearized equation. We show how these invariant manifolds can be used to extract information on higher-order long-time asymptotic expansions.

]]>Barycenter technique and two remaining questions in Geometric Analysis

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In this talk, we will present our recent solutions of the remaining cases of the boundary Yamabe problem and the Riemann mapping problem asked by Escobarin 1992. Rather than discussing our arguments of proofs, we will focus more on explaining the barycenter technique of Bahri-Coron which we employ. We hope by doing this to allow an easier understanding for the audience, since it seems to us, that even among experts, the barycenter technique is not known like the minimizing technique of Aubin-Schoen. Moreover, we hope also the audience to see how naturally the barycenter technique fits into conformally invariant variational problems verifying the structure of quantization and strong interaction phenomena.

(Joint work with M. Mayer of University of Giessen)

]]>Adams inequality on the Hyperbolic space

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In this talk we will discuss the classical Adams inequality and its versions in the hyperbolic space. We will also discuss the hyperbolic versions of Adachi-Tanaka type inequalities and the exact growth.

]]>On compactness of solution of the Quantum Navier-Stokes equations

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In this talk we focus on a new compactness result about weak solutions of the quantum Navier-Stokes equations. The novelty of the result is that we are able to consider the vacuum in the definition of weak solutions. The main tool is a new formulation of the equations which allows us to get an additional a priori estimate to prove compactness. Some remarks concerning the choice of the approximation system to get global existence will be made.

(Joint work with Paolo Antonelli)

]]>Compactness and stability issues for Einstein-Lichnerowicz constraints system

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I will survey recent results on the Einstein-Lichnerowicz constraints system which appears in general relativity when trying to formulate the Cauchy problem for the Einstein equation coupled with a scalar field. I will discuss existence, uniqueness,compactness and stability for this system. This is a joint work with Bruno Premoselli.

]]>Geometry in the Euler equations of hydrodynamics: h-principle and convex integration

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The following dichotomy between rigidity and flexibility is now well known in geometry: while uniqueness holds for smooth solutions to the isometric embedding problem, the set of solutions becomes unimaginably large if one allows rough ones. What is surprising is that this dichotomy holds for problems coming from mathematical physics,and in particular the Euler equations of fluid dynamics. In this (mainly expository) talk I will explain the h-principle and the method of convex integration. Convex geometry is the heart of the matter and profuse figures will attempt to illustrate the difficulties and how to tame them.

]]>Compactness Properties for Singular Liouville Equations and Systems

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I will give a brief overview of the main results concerning topological methods for singular Liouville equations on compact surfaces, and I will show how to extend some of them to special elliptic systems. My analysis will focus on sharp forms of the Moser-Trudinger inequality and on mass-quantization results for the SU(3) Toda System.

]]>Optimal trace ideals properties of the Fourier restriction operator and applications

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We study the trace ideals properties of the Fourier restriction operator to hypersurfaces. Equivalently, we generalize the theorems of Stein-Tomas and Strichartz to systems of orthonormal functions, with an optimal dependence on the number of such functions. As an application, we deduce new Strichartz inequalities describing the dispersive behaviour of the free evolution of quantum systems with an infinite number of particles. This is a joint work with Rupert Frank.

]]>Some recent results on the Jacobian equation

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Given two functions f and g,we want to find a map φ such that

g(φ(x)) det∇φ(x)=f(x) x∈Ω,

φ(x)=x x∈∂Ω.

Local case. We first consider the (local) existence, uniqueness and optimal regularity for the problem

g_{i}(φ(x)) det∇φ(x)=f_{i}(x) for every 1≤i≤n

where g_{i}·f_{i}>0.

*Global case*. A necessary condition is then

∫_{Ω} f =∫_{Ω} g. (1)

(i) We discuss the case where g·f>0 and give three different ideas for the existence problem with optimal regularity.

(ii) We then briefly comment on the case where g>0 but f is allowed to change sign.

*A problem without the condition (1)*. We consider a more general problem of the form

det∇φ(x)=f(x,φ(x),∇φ(x)) x∈Ω,

φ(x)=x x∈∂Ω.

where no constraint of the type (1) is needed.

]]>

A counter-example concerning regularity properties for systems of conservation laws

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In 1973 Schaeffer established a result that applies to scalar conservation laws with convex fluxes and can be loosely speaking formulated as follows: for a generic smooth initial datum, the admissible solution is smooth outside a locally finite number of curves in the (t,x) plane. Here the term ”generic” should be interpreted in a suitable technical sense, related to the Baire Category Theorem. My talk will aim at discussing a recent explicit counter-example that shows that Schaeffer’s Theorem does not extend to systems of conservation laws. The talk will be based on joint works with Laura Caravenna.

]]>Results and conjectures about some isoperimetric problems with density

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The standard isoperimetric inequality states that among all sets with a given fixed volume(or area in dimension 2) the ball has the smallest perimeter. That is, written here for simplicity in dimension 2, the following infimum is attained by the ball

2πR= inf{∫_{∂Ω} 1 dσ(x) : Ω⊂**R**^{2} and ∫_{Ω} 1 dx=πR^{2}}.

The isoperimetric problem with density is a generalization of this question: given two positive functions f,g:**R**^{2}→**R**^{2} one studies the existence of minimizers of

I(C) = inf{∫_{∂Ω} g(x) dσ(x) : Ω⊂**R**^{2} and ∫_{Ω} f(x) dx=C}.

I will mainly talk about the situation when f(x) =|x|q and g(x) =|x|p.This is a reach problem with strong variations in difficulties depending on the values of p and q. Some cases are still an open problem. One case has an interesting application related to the Moser-Trudinger imbedding. I will also mention the situation when f=g=eψ is strictly positive and radial, which leads to the log-convex density conjecture.

]]>A generalization of Gromov’s almost flat manifold theorem

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Almost flat manifolds are the solutions of bounded size perturbations of the equation Sec=0 (Sec is the sectional curvature). In a celebrated theorem, Gromov proved that the presence of an almost flat metric implies a precise topological description of the underlying manifold.

Integral pinching theorems express curvature assumptions in terms of certain L^{p}-norms and try to deduce topological conclusions. But typically one needs to require p >n2, where n is the dimension of the manifold, to prove such rigidity theorems.

During this talk we will explain how, under lower sectional curvature bounds, to imposeanL1-pinching condition on the curvature is surprisingly rigid, leading indeed to the same conclusion as in Gromov’s theorem under more relaxed curvature conditions.

This is a joint work with B. Wilking.

]]>Global solutions and asymptotics of Teichmüller harmonic map flow

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Teichmüller harmonic map flow is a gradient flow of the Dirichlet energy which is designed to evolve parametrized surfaces towards critical points of the area. In this talk we will discuss the construction and some new results for this flow and show in particular that for non-positively curved targets the flow changes or decomposes arbitrary closed initial surfaces into minimal immersions (possibly with branch points) through globally defined smooth solutions.

]]>Gibbs measures for the periodic derivative NLS equation

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The DNLS equation is an integrable PDE, in the sense that there are infinitely many Hamiltonians associated to it. The aim of the talk is to present the construction of infinitely many functional measures associated to these integrals of motion of the equation, each measure being supported on Sobolev spaces of increasing regularity. These are natural candidates to be the invariant measures associated to the DNLS eq. Invariant measures are a crucial tool in the theory of integrable PDEs, useful e.g. to prove long time properties of regular solutions. The introductory general aspects will be reviewed and the new results on DNLS, obtained in collaboration with R. Luc (ICMAT,Madrid) and D. Valeri (MSC, Beijing), will be presented.

]]>Stationary Kirchhoff systems in closed manifolds

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Area-minimizing graphs in the Heisenberg group

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We consider the area functional for graphs in the sub-Riemannian Heisenberg group and study minimizers of the associated Dirichlet problem. We prove that,under a bounded slope condition on the boundary datum, there exists a unique minimizer and that this minimizer is Lipschitz continuous. We also provide an example showing that, in the first Heisenberg group, Lipschitz regularity cannot be improved even under the bounded slope condition. This is based on a joint work with A. Pinamonti, F. SerraCassano and G. Treu.

]]>Frobenius property for integral currents and decomposition of normal currents

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In this joint work with Giovanni Alberti, we prove a Frobenius property for inte-gral currents: namely, if R=[∑,ξ,θ] k-dimensional integral current with a simple tangent vector field ξ∈C^{1}(**R**^{d};Λ_{k}(**R**^{d})), then ξ is involtive at almost every point in ∑. This result is related to the following decomposition problem formulated by F.Morgan: given a k-dimensional normal current T, do there exist a measure space L and a family of rectifiable currents {R_{λ}}_{λ∈L} such that T = ∫_{L} R_{λ} dλ and the mass decomposes consistently as M(T) = ∫_{L} M(R_{λ}) dλ? The aforementioned Frobenius property allows us to provide a counterexample to the existence of such a decomposition with a family of integral currents.

Artificial compressibility approximation of suitable weak solutions of incompressible Navier-Stokes equations

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In this talk I will discuss the problem of the approximation of suitable weak solutions of Navier-Stokes equations in the sense of Scheffer and Caffarelli-Kohn-Nirenberg. It is well-known that suitable weak solutions enjoy the partial regularity theorem proved in the famous paper of Caffarelli-Kohn-Nirenberg, hence they are more regular than a Leray weak solutions. However, since the uniqueness of weak solutions of Navier-Stokes is unknown we don’t know if different approximation methods lead to a suitable weak solution. I will present a recent result obtained with L. C. Berselli (University of Pisa) where we proved that weak solutions obtained by some artificial compressibility approximation are suitable. The novelty of the result is that the Navier-Stokes equations are considered in a bounded domain with Navier boundary conditions.

]]>The structure of complex unimodular maps

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We describe the structure of maps u:(0,1)^{n} → S^{1} having a given Sobolev regularity. Such maps are described by their singularities and phases. This is the analog of the Weierstrass factorization theorem for holomorphic functions; the singularities of the Sobolev maps play the role of the zeroes of holomorphic maps. We will present implications of this result to functional analytic questions related to manifold valued maps. If the time permits it, we will discuss the question of the control of the phases, and present some applications to some model PDEs and nonlocal problems.

Compactness issues and bubbling phenomena for the prescribed Gaussian curvature equation

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A classical question in differential geometry concerns which smooth functions f can arise as Gauss curvature of a conformal metric on a 2-dim Riemannian manifold M. This amounts to solve a PDE which is the Euler-Lagrange equation of an energy functional. In this talk we will discuss about compactness issues and bubbling phenomena for this equation on surfaces of genus greater than 1 (joint work with Borer and Struwe) and on the torus.

]]>On the Hardy-Schrödinger operator with a boundary singularity

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We investigate the Hardy-Schrödinger operator L_{γ}=-Δ-γ/|x|2 on domains Ω⊂**R**^{n}, whose boundary contain the singularity 0. The situation is quite different from the well-studied case when 0 is in the interior of Ω. For one, if 0∈Ω, then L is positive if and only if γ<(n-2)^{2}/4, while if 0∈∂Ω the operator L could be positive for larger value of γ, potentially reaching the maximal constant n^{2}/4 on convex domains.

We prove optimal regularity and a Hopf-type Lemma for variational solutions of corresponding linear Dirichlet boundary value problems of the form Lγ=a(x)u, but also for non-linear equations including Lγ=(|u|^{β-2}u)/(|x|^{s}), where γ < n^{2}/4, s∈[0,2) and β:=2(n-s)/(n-2) is the critical Hardy-Sobolev exponent. We also provide a Harnack inequality and a complete description of the profile of all positive solutions–variational or not– of the corresponding linear equation on the punctured domain. The value γ=(n-1)^{2}/4 turned out to be another critical threshold for the operator Lγ, and our analysis yields a corresponding notion of “Hardy singular boundary-mass” mγ(Ω) of a domain Ω having 0∈Ω, which could be defined whenever (n^{2}-1)/4 < γ < n^{2}/4.

As a byproduct, we give a complete answer to problems of existence of extremals for Hardy-Sobolev inequalities of the form

C( ∫_{Ω }(u^{β})/(|x|^{s}) dx )^{2/β }≤∫_{Ω} |∇u|^{2} dx - γ∫_{Ω} (u^{2})/(|x|^{s})dx

whenever γ<n^{2}/4, and in particular, for those of Caffarelli-Kohn-Nirenberg. These resultsextend previous contributions by the authors in the case γ=0, and by Chern-Lin for the case γ<(n-2)^{2}/4. Namely, if 0≤γ≤(n^{2}-1)/4, then the negativity of the mean curvature of ∂Ω at 0 is sucient for the existence of extremals. This is however not sufficient for (n^{2}-1)/4≤γ≤(n^{2})/4, which then requires the positivity of the Hardy singular boundary-massof the domain under consideration.

Joint work with Nassif Ghoussoub.

]]>Dissipative Hölder solutions to the incompressible Euler equations

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We consider the Cauchy problem for the incompressible Euler equations on the three-dimensional torus. According to a conjecture due to Onsager, which is well known in turbulence theory, while all the solutions which are uniformly α-Hölder continuous in space for any α>1/3 must conserve the total kinetic energy, for any α<1/3 there can be uniformly α-Hölder solutions which are strictly dissipative. While the first part of the conjecture is well established since a long time, the second part is still open in its full generality. In the result that we present we show that, for any α<1/5, there exist C^{α} vector fields being the initial data of infinitely many C^{α} solutions of the Euler equations which dissipate the total kinetic energy.

Properties of Coulombic eigenfunctions of atoms and molecules

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The eigenfunctions of the Schrödinger operator for (non-relativistic) atoms and molecules (in the Born-Oppenheimer/clamped nuclei approximation) are solutions of an elliptic partial differential equation with singular (total) potential (i.e., zero-order term). In this talk we give an overview over our results about the structure/regularity of the eigenfunctions at the singularities of the potential. These, in particular, improve on the well-known ’Kato Cusp Condition’. If time permits, we also discuss the implications for the electron density.

This is joint work with S. Fournais (Aarhus, Denmark), and M. and T. Hoffmann-Ostenhof (Vienna, Austria).

]]>Polynomial propagation of moments and global existence for a Vlasov-Poisson system with a point charge

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We consider the Cauchy problem associated to the Vlasov-Poisson system and we extend the well-posedness theory of Lions and Perthame to the case of initial data which include a Dirac mass. Moreover we provide polynomially growing in time estimates for the moments of the solution. This is a joint work with L. Desvillettes and E. Miot.

]]>Regularity of Free Boundaries in Anisotropic Capillarity Problems and the Validity of Young’s Law

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Local volume-constrained minimizers in anisotropic capillarity problems develop free boundaries on the walls of their containers. We prove the regularity of the free boundary outside a small set, showing in particular the validity of Young’s law at almost every point (joint with Francesco Maggi).

]]>Rayleigh-Bénard convection at finite Prandtl number:bounds on the Nusselt number

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We consider Rayleigh-Bénard convection at finite Prandtl number as modelled by the Boussinesq equation. We are interested in the scaling of the average upward heat transport, the Nusselt number Nu, in terms of the Rayleigh number Ra, and the Prandtl number Pr.

Physically motivated heuristics suggest the scaling Nu∼Ra^{1⁄3} and Nu∼Ra^{1/2} depending on Pr, in different regimes.

In this talk I present a rigorous upper bound for Nu reproducing both physical scalings in some parameter regimes up to logarithms. This is obtained by a (logarithmically failing) maximal regularity estimate inL1and inL1for the nonstationary Stokes equation with forcing term given by the buoyancy term and the nonlinear term, respectively. This is a joint work with Felix Otto and Antoine Choffrut.

]]>Singular Liouville systems and non-abelian Chern Simons vortices

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We discuss a class of singular Liouville systems in the plane and their role in the construction of non-abelian Chern-Simons vortices

]]>The Hartree equation for infinite quantum systems

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A Fermi gas occupying the whole euclidian space is an example of a translation-invariant quantum system with an infinite number of particles. We study its stability properties under the time-dependent nonlinear Hartree equation. If this system is slightly perturbed at the initial time, we show in particular that it returns to the translation-invariant state for large times. This is an instance of nonlinear dispersion for infinite quantum systems, which was recently studied by Frank, Lewin, Lieb and Seiringer in the linear case. This a joint work with Mathieu Lewin (CNRS/Cergy). I will also mention some recent work on Strichartz estimates for systems of orthonormal functions, joint with Rupert Frank (Caltech).

]]>Estimates for the topological degree and related problems

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In this talk, I first discuss estimates for the topological degree of maps from sphere into itself. Second, I present characterizations of Sobolev spaces based on the pointwise convergence or the Gamma-convergence of a sequence of nonlocal, nonconvex functionals related to these estimates. If time permits, I will also discuss the connection between these functionals with various filters in the denoising problem. The talk is based on joint works with Jean Bourgain and Haim Brezis.

]]>Integro-Differential harmonic maps into manifolds

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I will present results and ideas for the proof for regularity theory for critical points of non-local, degenerate integro-differential energies into manifolds which are related to p-harmonic maps.

]]>Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Following [1], in this talk we show how to establish, in a rather general setting, an analogue of DiPerna-Lions theory on well-posedness of flows of ODE’s associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into **R**^{∞}.

When specialized to the setting of Euclidean or infinite dimensional (e.g.Gaussian) spaces, large parts of previously known results are recovered at once.Moreover, the class of RCD(K,∞) metric measure spaces, recently introduced by Ambrosio, Gigli and Savar ́e, object of extensive recent research, fits into our framework. Therefore we provide, for the first time, well-posedness results forODE’s under low regularity assumptions on the velocity and in a non smooth context.

**References**:

[1] L. Ambrosio and D. Trevisan. Well posedness of Lagrangian flows and continuity equations in metric measure spaces. ArXiv e-prints, February 2014.

Unique Continuation for Fractional Schrödinger Equations with Rough Potentials

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This talk is focused on unique continuation principles for fractional Schrödinger equations with scaling-critical and rough potentials. The results are deduced via so-called Carleman estimates. In particular, these methods can be transferred to “variable coefficient” versions of fractional Schrödinger equations.

]]>Sufficient conditions for Willmore-immersions in R^3 to be minimal surfaces

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We provide two sharp sufficient conditions for immersed Willmore surfaces in **R**^{3}, definedon bounded C^{4}-subdomains of **R**^{2}, to be already minimal surfaces, i.e. to have vanishing mean curvatures on their entire domains. Our precise results read as follows:

**Theorem 1.** For some bounded C^{4}-domain Ω⊂**R**^{2} let X∈C^{4}(Ω,**R**^{3}) denote some immersed Willmore surface with Gauss map N and mean curvature H. Furthermore, assume that there exist constants c,d∈R and some fixed vector V∈S^{2} such that χ := cX+dV satisfies at least one of the following two conditions:

a) There is some “normal domain” G⊂Ω such that there hold H=0 on ∂G and H≥0 (or H≤0) in G∩O, where O⊂**R**^{2} is some open neighbourhood of ∂G, and

inf_{∂G}<χ,N> ≥ 0 as well as sup_{∂G}<χ,N> > 0;

b) H=0 on ∂Ω and

<χ,N> > 0 in Ω\A as well a sup_{∂Ω}<χ,N> > 0

for some finite set A⊂Ω.

Then H≡0 is satisfied in \bar{Ω}, i.e.X is a minimal surface on \bar{Ω}.

These results turn out to be particularly suitable for applications to Willmore graphs. We can therefore show that Willmore graphs on bounded C4 domains \bar{Ω} with vanishing mean curvatures on the boundary ∂Ω must already be minimal graphs. Our methods also prove that any closed Willmore surface in R3 which can be represented as a smooth graph over S2 has to be of constant, non-zero mean curvature and therefore a round sphere. Finally we demonstrate that our results are sharp by means of an examination of some certain part of the Clifford-Torus in R3.

]]>Incompressible Extensions in Sobolev Spaces

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Given a bounded domain and boundary data, does there exist a vector-valued map on this domain which is incompressible, that is, a map whose Jacobian determinant is one (almost) everywhere? In a regular setting, this question has been essentially positively answered in a famous paper by Dacorogna and Moser. I will present an analogous result in Sobolev spaces of low regularity, which was recently achieved by a convex integration method jointly with K. Koumatos (Oxford) and F. Rindler (Warwick). I will also comment on several generalisations and applications.

]]>Critical Mean Field Equations on multiply connected domains

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The uniqueness of solutions of the (Liouville) mean field-type equation on a simply connected domain and in the sub critical regime λ∈(0,8π) was first proved by T. Suzuki (1992). This result has been later improved by S.Y.A. Chang, C.C. Chen and C.S. Lin (2003) [CCL] to cover the critical value λ∈(0,8π]. The case where the domain is not simply connected has been a long-standing open problem which we have finally solved in a recent paper in collaboration with C.S. Lin. Our proof is based on a new generalization of a P.D.E. version of the Alexandrov-Bol's isoperimetric inequality on multiply connected domains. Another delicate problem is to understand the existence/non-existence of solutions for this equation on multiply connected domains at the critical parameterλ=8π. Criticality here means that the variational functional whose critical points are solutions of the equation is not anymore coercive for λ=8π, which implies in particular in this situation that existence/non existence of solutions depend on the geometry of the domain. I will discuss our generalization of a result in [CCL] which yield necessary and sufficient conditions for the existence of solutions for the mean field equation at the critical parameter λ=8π.

]]>Boundary regularity for elliptic integro-differential equations

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We study the boundary regularity of solutions to elliptic integro-differential equations. First we prove that, for the fractional Laplacian (-Δ)^{s} with s∈(0,1), solutions u satisfy that u/d^{s} is Hölder continuous up to the boundary, where d(x) is the distance to the boundary of the domain Ω. We will show that, in this fractional context, the quantity u/d^{s}|_{∂Ω} plays the role that the normal derivative plays in second order equations. Finally, we also present new boundary regularity results for fully nonlinear integro-differential equations.

On the chain rule for the divergence operator in $\mathbb{R}^2$

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Suppose that b:**R**^{d}**→R**^{d} is a vector field, β:**R****→R**^{ } is a smoth function and u:**R**^{2}**→R** is a scalar field. If both u and b are smooth then the following formula holds: div(β(u))b = β(u) - (β'(u) - uβ'(u)) div (b) + β'(u) div(ub). Generalizations of this formula when u∈L^{∞} and b belongs to Sobolev space or has bounded variation were studied by R. Di Perna, P.-L. Lions, L. Ambrosio, C. De Lellis, J. Maly and other authors. I will present a new result in this direction for d=2, which was obtained recently in collaboration with S. Bianchini. In particular our result holds when b is steady nearly incompressible BV vector field.

Existence of physical solutions to the semigeostrophic equations

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The semigeostrophic equations are a set of equations which model large-scale atmospheric/ocean flows.

The system admits a dual version, obtained from the original equations through a change of variable. Existence for the dual problem has been proven in 1998 by Benamou and Brenier, but the existence of a solution of the original system remained open due to the low regularity of the change of variable.

In the talk we prove the existence of distributional solutions of the original equations, both in R^{3} and in a two-dimensional periodic setting. The proof is based on recent regularity and stability estimates for Alexandrov solutions of the Monge-Ampère equation, established by De Philippis and Figalli.

The fractional Yamabe problem

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In the last years, a substantial amount of work has been devoted to understand elliptic, parabolic and hyperbolic problems with non local diffusion. In this talk, I will introduce a new class of conformally covariant operators of fractional order generalizing the scalar and Paneitz curvature. I will describe the associated Yamabe problem, in the regular and singular settings. I will give some existence results and discuss open problems.

]]>Measure-valued solutions of transport equations - where discrete meets continuous

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We consider a one-dimensional transport (balance) equation with velocity which has non-Lipschitz zeroes. This leads to non-uniqueness and concentration of characterics and dynamics with both discrete and continuous components. To deal with these effects, we use measure-valued solutions and the so-called measure-transmission conditions. A metric in the space of Radon measures allowing to define unique and stable solutions is introduced. The equation under consideration was proposed as a structured population model of cell differentiation.

]]>A Geometric Uncertainty Principle with an Application to Pleijel's estimate

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It is obvious that there is no tiling of the Euclidean plane with unit disks (any three disks have a gap in the middle): we prove a quantitative version of this statement. This simple insight has applications in spectral geometry: it tells us something about the topological structure of the vibration profile of a (possibly oddly-shaped) drum and allows us to recover an improved version of Pleijel's estimate (which was also recently done by Bourgain).

]]>Willmore spheres in Riemannian manifolds

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Given an immersion f of the 2-sphere in a Riemannian manifold (M,g) we study quadratic curvature functionals of the type: \int_{f(S^2)} H^2, \int_f(S^2) A^2, \int_{f(S^2)} )|Aº|^2, where H is the mean curvature, A is the second fundamental form, and Aº is the tracefree second fundamental form. Minimizers, and more generally critical points of such functionals can be seen respectively as GENERALIZED minimal, totally geodesic and totally umbilical immersions. In the seminar I will review some results (obtained in collaboration with Kuwert, Rivière and Shygulla) regarding the existence and the regularity of minimizers of such functionals. An interesting observation regarding the results obtained with Rivière is that the theory of Willmore surfaces can be usesfull to complete the theory of minimal surfaces (in particular in relation to the existence of canonical smooth representatives in homotopy classes, a classical program started by Sacks and Uhlenbeck).

]]>Rademacher theorem with respect to singular measures.

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Rademacher theorem states that every Lipschitz function on the euclidean space is differentiable almost everywhere with respect to the Lebesgue measure. In this talk I will explain how this statement should be modified when the Lebesgue measure is replaced by an arbitrary singular measure, and in particular I will show that the differentiability properties of Lipschitz functions with respect to such a measure are exactly described by the decompositions of the measure in terms of (measures on) rectifiable curves. This result is directly related to recent work by many authors, including myself, David Bate, Marianna Csornyei, Peter Jones, Andrea Marchese, and David Preiss.

]]>Nonlinear bounds states on manifolds

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We will discuss the results of several joint ongoing projects (with subsets of collaborators Pierre Albin, Hans Christianson, Colin Guillarmou, Jason Metcalfe, Laurent Thomann and Michael Taylor), which explore the existence, stability and dynamics of nonlinear bound states and quasimodes on manifolds of both positive and negative curvature with various symmetry properties.

]]>Enstrophy dissipation in 2D incompressible fluids

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I will present some (old) results on the transport and dissipation of enstrophy in 2D incompressible flows. Enstrophy is half the space integral of vorticity squared, and it is a relevant quantity in 2D turbulence. I consider initial data with vorticity in L^{2} and its logarithmic refinements and study exact transport of enstrophy by the velocity field. I also consider data in the larger Besov space $B^{0}_{2,\infty}$ and study the existence of well-defined enstrophy defects, measuring the rate of enstrophy dissipation.

This is joint work with Milton Lopes Fihlo and Helena Nussenzveig Lopes.

]]>Uniqueness of particle trajectories for incompressible fluids.

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We show some regularity results for some classes of 2D incompressible fluids, needed to show uniqueness of particle trajectories.

]]>Traffic flow modeling by conservation laws

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Several phenomena in traffic flow can be modeled through the use of conservation laws. We present two PDE-ODE coupled models that are used in different traffic situations. First, we consider a model that applies to moving bottlenecks and then we consider a model that applies in control problems for highway ramp metering. We provide a rigorous analytical framework for the Cauchy and Riemann problems and we show some numerical simulations.

]]>The role of anisotropy in the three-dimensional Navier-Stokes equations

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In this talk I will present some recent results in the study of the Cauchy problem for the three-dimensional Navier-Stokes equations. In particular using the fact that the two-dimensional equation is well-posed, I will try to explain the role of "spectral anisotropy" in the resolution of the equations.

]]>Blow up for critical wave equations on curved backgrounds.

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Following the work of Krieger, Schlag, and Tataru, we construct a family of blow-up solutions with finite energy norm to the equation

∂_{t}^{2}u - Δg u = |u|^{4} u.

This family has a continuous rate of blow up, but In contrast to the case where g is the Minkowski metric, the argument used to produce these solutions can only obtain blow up rates that are bounded above.

This is joint work with S. Shashahani.

]]>Regularity theory of degenerate elliptic equations in nondivergence form with applications to homogenization.

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I will present a regularity result for degenerate elliptic equations in nondivergence form. In joint work with Charlie Smart, we extend the regularity theory of Caffarelli to equations with possibly unbounded ellipticity - provided that the ellipticity satisfies an averaging condition. As an application we obtain a stochastic homogenization result for such equations which is equivalent to an invariance principle for random diffusions in random environments. The degenerate equations homogenize to uniformly elliptic equations, and we give an estimate of the ellipticity in terms of the averaging condition.

]]>Vortex sheets for 2D incompressible ideal fluids in domains with boundary

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Following the work of Krieger, Schlag, and Tataru, we construct a family of blow-up solutions with finite energy norm to the equation

∂_{t}^{2}u - Δg u = |u|^{4} u.

This family has a continuous rate of blow up, but In contrast to the case where g is the Minkowski metric, the argument used to produce these solutions can only obtain blow up rates that are bounded above.

This is joint work with S. Shashahani.

]]>When the Garding inequality is ineffective: examples and remedies

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The Garding inequality states that positive pseudo-differential symbols are associated with semi-positive operators. It can be used in particular to show time-exponential growth of solutions to initial value problems for elliptic equations. I will give examples in which Garding fails to give appropriate bounds, and a way to overcome this difficulty. Examples include high-frequency asymptotics of systems based on Maxwell's equations, and compressible Euler systems with a Van der Waals pressure law. In these cases, appropriate bounds are derived via a description of the parametrix of a pseudo-differential system.

]]>Regularity theory for area-minimizing currents

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A well-known theorem of Almgren shows that area-minimizing integral *k*-dimensional currents in a Riemannian manifold of arbitrary dimension* N* are regular up to a set of closed dimension of Hausdorff dimension at most *N-2*. In a joint work with Emanuele Spadaro we give a much shorter proof of this statement in the euclidean setting, following the general program of Almgren but introducing new ideas at the various steps. In this talk I will explain some if these ideas. A generalization of our proof to the Riemannian case is work in progress.

Spectral methods in the study of semi-linear equations in Hilbert spaces.

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I will present the results on the existence of solutions to semi-linear equation

Lx+N(x)=0,

where L is a linear and N a nonlinear operator defined on Hilbert space. I concentrate on the case when 0 is in an essential spectrum of L. The two main methods which I use are: topological degree in infinite-dimensional spaces and the spectral theory for linear operators in Hilbert spaces. This results are part of my Ph.D. project.

]]>Regularity issues for local minimizers of the Mumford-Shah functional in dimension two

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In this talk I shall focus on the higher integrability property enjoyed by the approximate gradients of local minimizers of the 2d Mumford-Shah energy. Related regularity issues shall be also discussed.

This is joint work with C. De Lellis (Universitaet Zuerich).

]]>Non-stadard solutions of the compressible Euler system

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The deceivingly simple–looking compressible Euler equations of gas dynamics have a long history of important contributions over more than two centuries. If we allow for discontinuous solutions, uniqueness and stability are lost. In order to restore such properties further restrictions on weak solutions have been proposed in the form of entropy inequalities. In this talk, we will discuss some counterexamples to the well–posedness theory of entropysolutions to the multi–dimensional compressible Euler equations. First, we show failure of uniqueness on a ﬁnite–time interval for entropy solutions starting from any continuously diﬀerentiable initial density and suitably constructed initial linear momenta. In other words, we prove that there exist wild initial data allowing for inﬁnitely many distinct entropy weak solutionsnof the compressible Euler system. Finally, we present a new upshot: a classical Riemann datum is a wild initial datum in 2 space–dimensions. All our methods are inspired by a new analysis of the incompressible Euler equations recently carried out by De Lellis and Székelyhidi and based on a revisited “h-principle”.

]]>Spectral stability estimates for the Laplace operator with either Dirichlet or Neumann boundary conditions

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The talk will focus on the eigenvalue problem for the Laplace operator defined in an open and bounded domain, with homogenous conditions of either Dirichlet or Neumann type assigned at the boundary. Under fairly weak regularity assumptions on the domain, the problem admits a diverging sequence of nonnegative eigenvalues. I will discuss some new quantitative estimates controlling how each of the eigenvalues change when the domain is perturbed. These estimates apply to Lipschitz and to so-called Reifenberg-flat domains. The proof is based on an abstract lemma which applies to both the Neumann and the Dirichlet problem and which could be applied to other classes of domains.

The talk will be based on joint works with A. Lemenant and E. Milakis.

]]>Antisymmetry and regularity

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Starting from the example of harmonic maps, we will find a class of PDE problems which enjoy an additional, at first glimpse hidden property: Antisymmetry! This feature enables us to deduce regularity assertions which heavily rely on Wente's theorem. For this latter, various approaches will be discussed. The presentation will be completed by a version of Wente's result for arbitrary dimension.

]]>The vanishing viscosity problem for the Navier-Stokes equations in bounded domains

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In this talk I will discuss the vanishing viscosity problem for the Navier-Stokes equations in a bounded domain. It is well-known that when Dirichlet conditions are imposed on the boundary the inviscid limit is currently an open and difficult problem. On the other hand when other type of boundary conditions are considered the situation became simpler. In this talk a particular type of Navier boundary conditions involving only the vorticity of the velocity field are considered. In particular, I will discuss recent results obtained in collaboration with Luigi Berselli (University of Pisa) concerning the inviscid limit in energy norm of the Leray weak solutions and the inviscid limit in higher norms of local smooth solutions of the Navier-Stokes equations.

]]>The Moser-Trudinger equation on a disk: blow-up behavior and non-existence

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We study the Moser-Trudinger equation Δu = λu Exp(u2), λ>0 on a 2-dimensional disk, arising from the Moser-Trudinger sharp embedding of H^{1}_{0}(Disk) into the Orlicz space of functions u with Exp(u2) integrable. We answer some long standing open questions:

a) The weak limit of a blowing-up sequence of solutions to the Moser-Trudinger equation on a disk is 0.

b) The Dirichlet energy of a blowing-up sequence of solutions on a disk converges to 4π.

c) For L large enough, the Moser-Trudinger equation on a disk admits no solution with Dirichlet energy larger than L.

This work is joint project with Andrea Malchiodi (SISSA - Trieste).

]]>Microscopic derivation of the Ginzburg-Landau equation

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We give the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof.

]]>Nonscattering solutions and blowup at infinity for the critical wave equation

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I present some recent results, obtained in collaboration with Joachim Krieger, on novel types of solutions to the critical wave equation in 3 spatial dimensions. These solutions either blow up at infinity or vanish at a prescribed rate. The existence of such exotic dynamics violates a strong version of the soliton resolution conjecture.

Francois Bouchut:

TBA

]]>Global existence and collisions for some configurations of nearly parallel vortex filaments

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A system of simplified equations has been derived by Klein, Majda and Damodaran to describe the dynamics of nearly parallel vortex filaments in incompressible 3D fluids. This system combines a 1D Schrödinger-type structure together with the 2D point vortex system. Global existence for small perturbations of exact parallel filaments has been established by Kenig, Ponce and Vega in the case of two filaments and for particular configurations of three filaments. In this talk I will present large time existence results for particular configurations of four filaments and for other particular configurations of N filaments for any N larger than 2. I will also discuss some situations of finite time filament collapse. This is joint work with Valeria Banica.

]]>Ground states for the nonlinear Schrödinger-equation with interface

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We are interested in ground states for the nonlinear Schrödinger-equation (NLS) with an interface between two purely periodic media. This means that the coefficients in the NLS model two different periodic media in each halfspace. The resulting problem no longer has a periodic structure. Using variational methods we give conditions on the coefficients such that ground states are created/prevented by the interface.

]]>Choptuik's critical spacetime exists

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About twenty years ago, Choptuik studied numerically the gravitational collapse (Einstein field equations) of a massless scalar field in spherical symmetry, and found strong evidence for a universal, self-similar solution at the threshold of black hole formation. We give a rigorous, computer assisted proof of the existence of Choptuik's spacetime, and show that it is real analytic. This is joint work with E. Trubowitz.

]]>Dimensional reduction of the optimal transport problem with convex norm costs

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We consider the optimal transportation problem with cost functions given by generic convex norms in **R**^{d} and absolutely continuous first marginals. We show the existence of a partition of **R**^{d} into k-dimensional sets, k=0,...,d, such that every optimal transport plan can be characterized, via disintegration of measures, as a family of optimal transport plans each moving a conditional probability of the first marginal inside one of these k-dimensional sets, along the directions of an extremal k-dimensional cone of the convex norm. Moreover, the conditional probabilities of the first marginal on these sets are absolutely continuous with respect to the k-dimensional Hausdorff measure on the k-dimensional sets on which they are concentrated, thus settling the longstanding Sudakov's problem of the existence of locally affine decompositions of **R**^{d} that reduce norm cost transportation problem to families of lower dimensional ones. Finally, due to the minimality of our partition with respect to this "dimensional reduction" property, applications to secondary cost functions obtained first minimizing with respect to a convex norm and then with respect to a finer one (e.g., a strictly convex one) will be shown. These results were obtained in collaboration with Stefano Bianchini (SISSA, Trieste).

An introduction to (partial) regularity theory for elliptic problems

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In this seminar we will give a survey on some aspects of the classical regularity theory for W^{1,p}-solutions to elliptic problems (convex variational integral or elliptic systems), restricting ourselves to simple model cases and explaining the challenges behind proving such results. For scalar valued solutions full regularity (continuous or even better) can be established under very mild assumptions, which is nowadays known as the De Giorgi-Nash-Moser theory. In the vectorial case instead, the various component functions and their partial derivative can interact in such a way that the system or variational integral under consideration allows discontinuous or even unbounded solutions, and in fact various counterexamples to full regularity have been constructed. As a consequence, only partial regularity can be expected, in the sense that the solution (or its gradient) is locally continuous outside of a negligible set (the singular set). We will give some heuristics on the general

approach to partial regularity results and then we briefly discuss how in some particular situations (small space dimensions, special structure conditions) an upper bound on the Hausdorff dimension of the singular set can be obtained.