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Wei-Xi Li

Publications and source records attributed to Wei-Xi Li.

At least 37 records · Page 2Linked to original sources
arXiv2020

Well-posedness in Gevrey function space for 3D Prandtl equations without Structural Assumption

We establish the well-posedness in Gevrey function space with optimal class of regularity 2 for the three dimensional Prandtl system without any structural assumption. The proof combines in a novel way a new cancellation in the system with some of the old ideas to overcome the difficulty of the loss of derivatives in the system.This shows that the three dimensional instabilities in the system leading to ill-posedness are not worse than the two dimensional ones.

math.AP↗
arXiv2019

Well-posedness in Gevrey function space for the three-dimensional Prandtl equations

In the paper, we study the three-dimensional Prandtl equations without any monotonicity condition on the velocity field. We prove that when one tangential component of the velocity field has a single curve of non-degenerate critical points with respect to the normal variable, the system is locally well-posed in the Gevrey function space with Gevrey index in $]1, 2].$ The proof is based on some new observation of cancellation mechanism in the three space dimensional system in addition to those in the two-dimensional setting obtained in [1,7,19,22].

math.AP↗
arXiv2018

Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff

In this article we provide global subelliptic estimates for the linearized inhomogeneous Boltzmann equation without angular cutoff, and show that some global gain in the spatial direction is available although the corresponding operator is not elliptic in this direction. The proof is based on a multiplier method and the so-called Wick quantization, together with a careful analysis of the symbolic properties of the Weyl symbol of the Boltzmann collision operator.

math.AP↗
arXiv2018

Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cutoff. This equation is partially elliptic in the velocity direction and degenerates in the spatial variable. We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with Gevrey index depending on the angular singularity. Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of linearized Boltzmann operator.

math.AP↗
arXiv2018

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, 1985].

math.AP↗
arXiv2017

Well-posedness in Gevrey space for the Prandtl equations with non-degenerate critical points

In the paper, we study the Prandtl system with initial data admitting non-degenerate critical points. For any index $σ\in[3/2, 2],$ we obtain the local in time well-posedness in the space of Gevrey class $G^σ$ in the tangential variable and Sobolev class in the normal variable so that the monotonicity condition on the tangential velocity is not needed to overcome the loss of tangential derivative. This answers the open question raised in the paper of D. Gérard-Varet and N. Masmoudi [{\it Ann. Sci. Éc. Norm. Supér}. (4) 48 (2015), no. 6, 1273-1325], in which the case $σ=7/4$ is solved.

math.AP↗
arXiv2017

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.

math.AP↗
arXiv2017

Weighted gevrey class regularity of euler equation in the whole space

In this paper we study the weighted Gevrey class regularity of Euler equation in the whole space R 3. We first establish the local existence of Euler equation in weighted Sobolev space, then obtain the weighted Gevrey regularity of Euler equation. We will use the weighted Sobolev-Gevrey space method to obtain the results of Gevrey regularity of Euler equation, and the use of the property of singular operator in the estimate of the pressure term is the improvement of our work.

math.AP↗
arXiv2017

Vanishing viscosity limit of navier-stokes equations in gevrey class

In this paper we consider the inviscid limit for the periodic solutions to Navier-Stokes equation in the framework of Gevrey class. It is shown that the lifespan for the solutions to Navier-Stokes equation is independent of viscosity, and that the solutions of the Navier-Stokes equation converge to that of Euler equation in Gevrey class as the viscosity tends to zero. Moreover the convergence rate in Gevrey class is presented.

math.AP↗
arXiv2016

Gevrey regularity with weight for incompressible Euler equation in the half plane

In this work we prove the weighted Gevrey regularity of solutions to the incompressible Euler equation with initial data decaying polynomially at infinity. This is motivated by the well-posedness problem of vertical boundary layer equation for fast rotating fluid. The method presented here is based on the basic weighted $L^2$- estimate, and the main difficulty arises from the estimate on the pressure term due to the appearance of weight function.

math.AP↗
arXiv2016

Boundary Layer Analysis for the Fast Horizontal Rotating Fluids

It is well known that, for fast rotating fluids with the axis of rotation being perpendicular to the boundary, the boundary layer is of Ekman-type, described by a linear ODE system. In this paper we consider fast rotating fluids, with the axis of rotation being parallel to the boundary. We show that the corresponding boundary layer is describe by a nonlinear, degenerated PDE system which is similar to the 2-D Prandtl system. Finally, we prove the well-posedness of the governing system of the boundary layer in the space of analytic functions with respect to tangential variable.

math.AP↗
arXiv2016

Compactness Criteria for the Resolvent of the Fokker-Planck operator

In this paper we study the spectral property of a Fokker-Planck operator with potential. By virtue of a multiplier method inspired by Nicolas Lerner, we obtain new compactness criteria for its resolvent, involving the control of the positive eigenvalues of the Hessian matrix of the potential.

math.AP↗
arXiv2012

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 admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index $s$ equals to 1, then we obtain analyticity of the stream function. The regularity results hold for both periodic and solitary water waves.

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arXiv2011

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 [4] to the non-parabolic general case here.

math.AP↗
arXiv2010

Global Hypoellipticity and Compactness of Resolvent for Fokker-Planck Operator

In this paper we study the Fokker-Planck operator with potential V(x), and analyze some kind of conditions imposed on the potential to ensure the validity of global hypoelliptic estimates. As a consequence, we obtain the compactness of resolvent of the Fokker-Planck operator if either the Witten Laplacian on 0-forms has a compact resolvent or some additional assumption on the behavior of the potential at infinity is fulfilled. This work improves the previous results of Hérau-Nier and Helffer-Nier, by obtaining a better global hypoelliptic estimate under weaker assumptions on the potential.

math.AP↗
arXiv2009

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.

math.AP↗