Semi classical limit for a NLS with potential

This paper is dedicated to the semiclassical limit of t the nonlinear focusing Schrödinger equation (NLS) with a potential , i\e\partial_t u^{\e}+\frac{\e^2}{2}\lap u^{\e}-V(x)u^{\e}+|u^{\e}|^{2\sigma}u^{\e}=0 with initial data in the form Q\left(\frac{x-x_0}{\e}\right)e^{i\frac{x.v_0}{\e}}, where $Q$ is the ground state of the associated unscaled elliptic problem. Using a refined version of the method introduced in \cite{BJ} by J. C. Bronski, R.L. Jerrard, we prove that, up to a time-dependent phase shift, the initial shape is conserved with parameters that are transported by the classical flow of the classical Hamiltonian $H(t,x)=\frac{|\xi|^2}{2}+V( x)$. This gives, in particular, a complete description of the dynamics of the time-dependent Wigner measure associated to the family of solutions

On the global existence for the axisymmetric Euler equations

This paper deals with the global well-posedness of the 3D axisymmetric Euler equations for initial data lying in some critical Besov spacesComment: 14 page

Limite non visqueuse pour le système de Navier-Stokes dans un espace critique

International audienceDans un article récent [11], Vishik montre que le système d'Euler bidimensionnel est globalement bien posé dans l'espace de Besov critique $B^2_{2,1}$. Nous montrons ici que le système de Navier-Stokes est globalement bien posé dans $B^2_{2,1}$, avec des estimations uniformes par rapport à la viscosité. Nous prouvons également un résultat global de limite non visqueuse. Le taux de convergence dans $L^2$ est de l'ordre $\nu$

Energy scattering for a class of inhomogeneous biharmonic nonlinear Schr\"odinger equations in low dimensions

We consider a class of biharmonic nonlinear Schr\"odinger equations with a focusing inhomogeneous power-type nonlinearity $$i\partial_t u -\Delta^2 u+\mu\Delta u +|x|^{-b} |u|^\alpha u=0, \quad \left. u\right|_{t=0}=u_0 \in H^2(\mathbb{R}^d)$$ with $d\geq 1, \mu\geq 0$, $00$, and $\alpha<\frac{8-2b}{d-4}$ if $d\geq 5$. We first determine a region in which solutions to the equation exist globally in time. We then show that these global-in-time solutions scatter in $H^2(\mathbb{R}^d)$ in three and higher dimensions. In the case of no harmonic perturbation, i.e., $\mu=0$, our result extends the energy scattering proved by Saanouni [Calc. Var. 60 (2021), art. no. 113] and Campos and Guzm\'an [Calc. Var. 61 (2022), art. no. 156] to three and four dimensions. Our energy scattering is new in the presence of a repulsive harmonic perturbation $\mu>0$. The proofs rely on estimates in Lorentz spaces which are properly suited for handling the weight $|x|^{-b}$