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Wei-Xi Li

Publications and source records attributed to Wei-Xi Li.

At least 19 recordsLinked to original sources
arXiv2026

Local well-posedness for the Boltzmann equation with hard potentials

We consider the spatially inhomogeneous non-cutoff Boltzmann equation with hard potentials in the non-perturbative setting. For initial data with polynomial decay in the velocity variable, we establish the local-in-time existence and uniqueness of weak solutions, conditional to pointwise bounds on the hydrodynamic quantities (mass, energy, and entropy). Compared to the soft potential case, the key challenge for full-range hard potentials lies in the more severe loss of velocity moments. The proof combines a hypoelliptic estimate with interpolation inequalities to handle the moment-loss terms.

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arXiv2026

On the radius of analyticity and Gevrey regularity for the Boltzmann equation

This paper investigates the non-cutoff Boltzmann equation for hard potentials in a perturbative setting. We first establish a sharp short-time estimate on the radius of analyticity and Gevrey regularity of mild solutions. Furthermore, we obtain a global-in-time radius estimate in Gevrey space. The proof combines hypoelliptic estimates with the macro-micro decomposition.

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arXiv2025

Gevrey well-posedness of the hydrostatic MHD-wave system

This paper investigates the well-posedness of the hydrostatic MHD-wave system. Unlike the standard hydrostatic MHD equations, the tangential magnetic field equation in this system is degenerate hyperbolic rather than parabolic, which leads to substantial mathematical difficulties. Using the boundary decomposition method, we establish local well-posedness in Gevrey $\frac{7}{6}$ space for convex initial data.

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arXiv2025

Global Gevrey solution of 3D anisotropic Navier-Stokes system in a strip domain

We investigate the three-dimensional (3D) incompressible anisotropic Navier-Stokes system with dissipation only in the horizontal variables, posed in a strip domain. To overcome the difficulties arising from the boundary terms and the absence of vertical dissipation, we impose a Gevrey-class regularity condition in the vertical direction. For the remaining directions, we prove that the solution exhibits space-time analytic or Gevrey-class regularization. Furthermore, the solution is shown to possess an enhanced Gevrey regularity in the direction of strong diffusion, which is unconstrained by boundaries.

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arXiv2025

Global Well-Posedness for the 2D and 3D Prandtl-Shercliff Model

We investigate the Prandtl-Shercliff model in both two and three dimensions. For the two-dimensional case, we establish global-in-time well-posedness in Sobolev spaces without any structural assumptions on the initial data. Furthermore, we show that the solution exhibits an analytic regularization effect in all variables, which holds globally in time and in space up to the boundary. For the three-dimensional case, we study a linearized version of the model and prove its global-in-time well-posedness for initial data that are analytic in only one tangential direction. The proofs rely crucially on the intrinsic non-local diffusion induced by the Shercliff boundary layer.

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arXiv2024

On the Boltzmann equation with soft potentials: Existence, uniqueness and smoothing effect of mild solutions

We consider the spatially inhomogeneous Boltzmann equation without angular cutoff for soft potentials. For any given initial datum such that the mass, energy and entropy densities are bounded and the mass is away from vacuum, we establish the local-in-time existence and uniqueness of mild solutions, and further provide the first result on sharp smoothing effect in analytic space or Gevrey space for soft potentials.

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arXiv2024

Global well-posedness of the MHD boundary layer equations in the Sobolev Space

We study the two-dimensional MHD boundary layer equations. For small perturbation around a tangential background magnetic field, we obtain the global-in-time existence and uniqueness of solutions in Sobolev spaces. The proof relies on the novel combination of the well-explored cancellation mechanism and the idea of linearly-good unknowns, and we use the former idea to deal with the top tangential derivatives and the latter one admitting fast decay rate to control lower-order derivatives.

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arXiv2024

Gevrey well-posedness of quasi-linear hyperbolic Prandtl equations

We study the hyperbolic version of the Prandtl system derived from the hyperbolic Navier-Stokes system with no-slip boundary condition. Compared to the classical Prandtl system, the quasi-linear terms in the hyperbolic Prandtl equation leads to an additional instability mechanism. To overcome the loss of derivatives in all directions in the quasi-linear term, we introduce a new auxiliary function for the well-posedness of the system in an anisotropic Gevrey space which is Gevrey class $\frac 32$ in the tangential variable and is analytic in the normal variable.

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arXiv2024

Sharp regularization effect for the non-cutoff Boltzmann equation with hard potentials

For the Maxwellian molecules or hard potentials case, we verify the smoothing effect for the spatially inhomogeneous Boltzmann equation without angular cutoff. Given initial data with low regularity, we prove its solutions at any positive time are analytic for strong angular singularity, and in Gevrey class with optimal index for mild angular singularity. To overcome the degeneracy in the spatial variable, a family of well-chosen vector fields with time-dependent coefficients will play a crucial role, and the sharp regularization effect of weak solutions relies on a quantitative estimate on directional derivatives in these vector fields.

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arXiv2022

Global Well-posedness of a Prandtl Model from MHD in Gevrey Function Spaces

We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer. A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index $2$. The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator

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arXiv2022

Analytic smoothing effect of the spatially inhomogeneous Landau equations for hard potentials

We study the spatially inhomogeneous Landau equations with hard potential in the perturbation setting, and establish the analytic smoothing effect in both spatial and velocity variables for a class of low-regularity weak solutions. This shows the Landau equations behave essentially as the hypoelliptic Fokker-Planck operators. The spatial analyticity relies on a new time-average operator, and the proof is based on a straightforward energy estimate with a careful estimate on the derivatives with respect to the new time-average operator.

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arXiv2022

3D hyperbolic Navier-Stokes equations in a thin strip: global well-posedness and hydrostatic limit in Gevrey space

We consider the hyperbolic version of three-dimensional anisotropic Naver-Stokes equations in a thin strip and its hydrostatic limit that is a hyperbolic Prandtl type equations. We prove the global-in-time existence and uniqueness for the two systems and the hydrostatic limit when the initial data belong to the Gevrey function space with index 2. The proof is based on a direct energy method by observing the damping effect in the systems.

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arXiv2021

Gevrey well-posedness of the hyperbolic Prandtl equations

We study the 2D and 3D Prandtl equations of degenerate hyperbolic type, and establish without any structural assumption the Gevrey well-posedness with Gevrey index $\leq 2$. Compared with the classical parabolic Prandtl equations, the loss of the derivatives, caused by the hyperbolic feature coupled with the degeneracy, can't be overcame by virtue of the classical cancellation mechanism that developed for the parabolic counterpart. Inspired by the abstract Cauchy-Kowalewski theorem and by virtue of the hyperbolic feature, we give in this text a straightforward proof, basing on an elementary $L^2$ energy estimate. In particular our argument does not involve the cancellation mechanism used efficiently for the classical Prandtl equations.

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arXiv2021

Well-posedness in Sobolev spaces of the two-dimensional MHD Boundary layer equations without viscosity

We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields dominates. This gives a complement to the previous works of Liu-Xie-Yang [Comm. Pure Appl. Math. 72 (2019)] and Liu-Wang-Xie-Yang [J. Funct. Anal. 279 (2020)], where the well-posedness theory was established for the MHD boundary layer systems with both viscosity and resistivity and with viscosity only, respectively. We use the pseudo-differential calculation, to overcome a new difficulty arising from the treatment of boundary integrals due to the absence of the diffusion property for the velocity.

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arXiv2021

Gevrey regularity of mild solutions to the non-cutoff Boltzmann equation

In the paper, for the Cauchy problem on the non-cutoff Boltzmann equation in torus, we establish the global-in-time Gevrey smoothness in velocity and space variables for a class of low-regularity mild solutions near Maxwellians with the Gevrey index depending only on the angular singularity. This together with [24] provides a self-contained well-posedness theory for both existence and regularity of global solutions for initial data of low regularity in the framework of perturbations. For the proof we treat in a subtle way the commutator between the regularization operators and the Boltzmann collision operator involving rough coefficients, and this enables us to combine the classical Hörmander's hypoelliptic techniques together with the global symbolic calculus established for the linearized Boltzmann operator so as to improve the regularity of solutions at positive time.

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arXiv2020

Well-posedness of the MHD boundary layer system in Gevrey function space without Structural Assumption

We establish the well-posedness of the MHD boundary layer system in Gevrey function space without any structural assumption. Compared to the classical Prandtl equation, the loss of tangential derivative comes from both the velocity and magnetic fields that are coupled with each other. By observing a new type of cancellation mechanism in the system for overcoming the loss derivative degeneracy, we show that the MHD boundary layer system is well-posed with Gevrey index up to $3/2$ in both two and three dimensional spaces.

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