Wei-Xi Li

Gevrey hypoellipticity for a class of kinetic equations

In this paper, we study the Gevrey regularity of weak solutions for a class of linear and semi-linear kinetic equations, which are the linear model of spatially inhomogeneous Boltzmann equations without an angular cutoff.

Gevrey Class Smoothing Effect for the Prandtl Equation

It is well known that the Prandtl boundary layer equation is unstable for general initial data, and is well-posed in Sobolev space for monotonic initial data. Recently, under the Oleiniks monotonicity assumption for the initial datum, R. Alexandre, Y. Wang, C.-J. Xu, and T. Yang [J. Amer. Math. Soc., 28 (2015) pp. 745--784] recovered the local well-posedness of the Cauchy problem in Sobolev space by virtue of an energy method (see also N. Masmoudi and T. K. Wong [Comm. Pure Appl. Math., 68 (2015), pp. 1683--1741.]). In this work, we study the Gevrey smoothing effects of the local solution obtained in R. Alexandre, Y. Wang, C.-J. Xu, and T. Yang [J. Amer. Math. Soc., 28 (2015) pp. 745--784]. We prove that the Sobolevs class solution belongs to some Gevrey class with respect to tangential variables at any positive time.

Regularity of Traveling Free Surface Water Waves with Vorticity

We prove real analyticity of all the streamlines, including the free surface, of a gravity- or capillary-gravity-driven steady flow of water over a flat bed, with a Hölder continuous vorticity function, provided that the propagating speed of the wave on the free surface exceeds the horizontal fluid velocity throughout the flow. Furthermore, if the vorticity possesses some Gevrey regularity of index s, then the stream function of class C2,μ admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index s equals 1, then we obtain analyticity of the stream function. The regularity results hold not only for periodic or solitary-water waves, but also for any solution to the hydrodynamic equations of class C2,μ.

Global hypoelliptic estimates for Landau-type operators with external potential

In this paper we study a Landau-type operator with an external force. It is a linear model of the Landau equation near Maxwellian distributions. Making use of multiplier method, we get the global hypoelliptic estimate under suitable assumptions on the external potential.

Hypocoercivity for the Linear Boltzmann Equation with Confining Forces

This paper is concerned with the hypercoercivity property of solutions to the Cauchy problem on the linear Boltzmann equation with a confining potential force. We obtain the exponential time rate of solutions converging to the steady state under some conditions on both initial data and the potential function. Specifically, initial data is properly chosen such that the conservation laws of mass, total energy and possible partial angular momentums are satisfied for all nonnegative time, and a large class of potentials including some polynomials are allowed. The result also extends the case of parabolic forces considered in Duan (Nonlinearity 24(8):2165–2189 (2011)) to the non-parabolic general case here.

Global Gevrey hypoellipticity for the twisted Laplacian on forms

We study in this paper the global hypoellipticity property in the Gevrey category for the generalized twisted Laplacian on forms. Different from the 0-form case, where the twisted Laplacian is a scalar operator, this is a system of differential operators when acting on forms, each component operator being elliptic locally and degenerate globally. We obtain here the global hypoellipticity in anisotropic Gevrey space.

Compactness of the Resolvent for the Witten Laplacian

In this paper, we consider the Witten Laplacian on 0-forms and give sufficient conditions under which the Witten Laplacian admits a compact resolvent. These conditions are imposed on the potential itself, involving the control of high-order derivatives by lower ones, as well as the control of the positive eigenvalues of the Hessian matrix. This compactness criterion for resolvent is inspired by the one for the Fokker–Planck operator. Our method relies on the nilpotent group techniques developed by Helffer–Nourrigat (Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Birkhäuser Boston Inc., Boston, 1985).