Asymptotic expansion of low-energy excitations for weakly interacting bosons]]>

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).

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

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. ]]>

Optimal regularity for viscous Hamilton-Jacobi equations in Lebesgue spaces]]>