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

Abstract

We consider the dispersive logarithmic Schrödinger equation in a semiclassical scaling. We extend the results of Carles and Gallagher (Duke Math. J. 167:9 (2018), 1761–1801) about the large-time behavior of the solution (dispersion faster than usual with an additional logarithmic factor and convergence of the rescaled modulus of the solution to a universal Gaussian profile) to the case with semiclassical constant. We also provide a sharp convergence rate to the Gaussian profile in the 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 behavior 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.