43 research outputs found
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 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 . 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 . Nous montrons ici que le système de Navier-Stokes est globalement bien posé dans , avec des estimations uniformes par rapport à la viscosité. Nous prouvons également un résultat global de limite non visqueuse. Le taux de convergence dans est de l'ordre
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 with , ,
and if . 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 in three and higher
dimensions. In the case of no harmonic perturbation, i.e., , 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 . The proofs rely on estimates in
Lorentz spaces which are properly suited for handling the weight