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.

我们考虑半经典缩放中的色散对数Schrödinger方程。我们扩展了Carles和Gallagher的结果（Duke Math。J. 167：9（2018），1761–1801）关于具有半经典常数的情况的解决方案（具有额外对数因子的色散比平常更快，并且解决方案的缩放比例模量收敛到通用高斯分布） 。通过对由该模量满足的Fokker-Planck方程的详细分析，我们还为Kantorovich-Rubinstein度量中的高斯分布提供了急剧的收敛速度。此外，由于有了Wigner变换，我们才对该方程进行了半经典极限运算，从而获得（Wigner）度量。我们证明这两个特征是兼容的，并且维格纳测度的密度与对数薛定ding方程解的模量具有相同的长时间行为。最后，我们讨论有关的动力学方程（即动力学等温欧拉系统）及其形式性质，受到先前结果和一类新的显式解法的启发。