The Wigner-Poisson-Fokker-Planck system: global-in-time solution and dispersive estimates

Wigner measures and observability for the Schrödinger equation on the disk

We analyse the structure of semiclassical and microlocal Wigner measures for solutions to the linear Schrödinger equation on the disk, with Dirichlet boundary conditions. Our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk. We prove a "structure theorem", expressing the restriction of the Wigner measures on each invariant torus in terms of {\em second-microlocal measures}. They are obtained by performing a finer localization in phase space around each of these tori, at the limit of the uncertainty princip…

High-frequency propagation for the Schroedinger equation on the torus

The main objective of this paper is understanding the propagation laws obeyed by high-frequency limits of Wigner distributions associated to solutions to the Schroedinger equation on the standard d-dimensional torus T^{d}. From the point of view of semiclassical analysis, our setting corresponds to performing the semiclassical limit at times of order 1/h, as the characteristic wave-length h of the initial data tends to zero. It turns out that, in spite that for fixed h every Wigner distribution satisfies a Liouville equation, their limits are no longer uniquely determined by those of the Wigne…

Local well-posedness for Boltzmann's equation and the Boltzmann hierarchy via Wigner transform

We use the dispersive properties of the linear Schrödinger equation to prove local well-posedness results for the Boltzmann equation and the related Boltzmann hierarchy, set in the spatial domain $\mathbb{R}^d$ for $d\geq 2$. The proofs are based on the use of the (inverse) Wigner transform along with the spacetime Fourier transform. The norms for the initial data $f_0$ are weighted versions of the Sobolev spaces $L^2_v H^α_x$ with $α\in \left( \frac{d-1}{2},\infty\right)$. Our main results are local well-posedness for the Boltzmann equation for cutoff Maxwell molecules and hard spheres, as we…

非线性量子Fokker-Planck方程的不可压缩Navier-Stokes-Fourier极限

Incompressible Navier-Stokes-Fourier limit from a nonlinear quantum Fokker-Planck equation

该研究从带玻色-爱因斯坦或费米-狄拉克统计的非线性量子Fokker-Planck方程，推导得到保留量子统计效应的不可压缩Navier-Stokes-Fourier极限，明确了相关模式的紧性与声学模式的色散消失特性。

Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schr{ö}dinger equation

We consider the dispersive logarithmic Schr{ö}dinger 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…

Semiclassical Completely Integrable Systems : Long-Time Dynamics And Observability Via Two-Microlocal Wigner Measures

We look at the long-time behaviour of solutions to a semi-classical Schrödinger equation on the torus. We consider time scales which go to infinity when the semi-classical parameter goes to zero and we associate with each time-scale the set of semi-classical measures associated with all possible choices of initial data. On each classical invariant torus, the structure of semi-classical measures is described in terms of two-microlocal measures, obeying explicit propagation laws. We apply this construction in two directions. We first analyse the regularity of semi-classical measures, and we emph…

论非线性 Schrödinger 流对非线性项幂次的依赖性

On the dependence of the nonlinear Schrodinger flow upon the power of the nonlinearity

本文研究散焦能量次临界幂型 NLS 流随非线性幂次的连续性，建立局部/全局时间连续、重标度模平方的一致收敛及幂次趋于零时向对数 NLS 的收敛。

Boltzmann方程的矩和正则性：通过Wigner变换

Moments and Regularity for a Boltzmann Equation via Wigner Transform

本文通过Wigner变换工具研究Boltzmann方程的解的矩和正则性，证明了在特定条件下解的非负性、能量守恒和H定理，以及Sobolev空间中的正则性保持。

Universal dynamics for the defocusing logarithmic Schrodinger equation

We consider the nonlinear Schrodinger equation with a logarithmic nonlinearity in a dispersive regime. We show that the presence of the nonlinearity affects the large time behavior of the solution: the dispersion is faster than usual by a logarithmic factor in time and the positive Sobolev norms of the solution grow logarithmically in time. Moreover, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile. These properties are suggested by explicit computations in the case of Gaussian initial data, and remain when an extra power-li…

薛定谔-克莱因-戈登系统再探

The Schrödinger-Klein-Gordon System Revisited

本文通过傅里叶-维格纳变换将薛定谔-克莱因-戈登系统重写为相空间上的非线性傅里叶-穆瓦亚尔方程，建立适定性与全局解，为汤川耦合提供微局域分析新框架。

具有一般Lindbladian的开放量子系统中超出埃伦费斯特时间的经典对应

Classical correspondence beyond the Ehrenfest time for open quantum systems with general Lindbladians

该研究证明了带一般Lindbladian的开放量子系统中，当扩散强度满足D≫ℏ^{4/3}时，量子-经典对应可超出埃伦费斯特时间，采用可变压缩高斯态半经典混合方法，论证了4/3指数的最优性，并在姊妹篇给出物理系统可直接应用的显式界。