Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation

Guillaume Ferriere (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

We consider the dispersive logarithmic Schrödinger equation in a semi-classical scaling. We extend the results about the large time behaviour of the solution (dispersion faster than usual with an additional logarithmic factor, convergence of the rescaled modulus of the solution to a universal Gaussian profile) to the case with semi-classical constant. We also provide a sharp convergence rate to the Gaussian profile in Kantorovich-Rubinstein metric through a detailed analysis of the Fokker-Planck equation satisfied by this modulus. Moreover, we perform the semiclassical limit of this equation thanks to the Wigner Transform in order to get a (Wigner) measure. We show that those two features are compatible and the density of a Wigner Measure has the same large time behaviour as the modulus of the solution of the logarithmic Schrödinger equation. Lastly, we discuss about the related kinetic equation (which is the Kinetic Isothermal Euler System) and its formal properties, enlightened by the previous results and a new class of explicit solutions.

Mots clés: Logarithmic Schrödinger equation; Large time behaviour; Semiclassical limit; Wigner Measures; Wasserstein distances; Harmonic Fokker-Planck operator; Kinetic Isothermal Euler System

Domaines: Equations aux dérivées partielles [math.AP]

Guillaume Ferriere. Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation. Analysis & PDE, 2021, 14 (2), pp.617-666. ⟨10.2140/apde.2021.14.617⟩. ⟨hal-02061983v2⟩