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Preprints

- Wei-Xi Li and
Tong Yang

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

– In the paper, we study the three-dimensional Prandtl equations, and prove that if one component of the tangential velocity field satisfies the monotonicity assumption in the normal direction, then the system is locally well-posed in the Gevrey function space with Gevrey index in ]1, 2]. The proof relies on some new cancellation mechanism in the system in addition to those observed in the two-dimensional setting.arXiv:1708.08217 - Wei-Xi Li, Alberto
Parmeggiani and Yan-Lin Wang

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.arXiv:1708.03095 - Wei-Xi Li

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].arXiv:1707.04745 - Wei-Xi Li,Van-Sang
Ngo and
Chao-Jiang Xu

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.arXiv:1611.04896 - Radjesvarane
Alexandre, Frédéric
Hérau and Wei-Xi Li

Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff.– In this article we provide global subelliptic estimates for the linearized inhomoge- neous 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.arXiv:1212.4632

- Wei-Xi Li and
Tong Yang

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

Accepted by Journal of the European Mathematical Society (JEMS) - Feng Cheng, Wei-Xi Li and
Chao-Jiang Xu

Vanishing viscosity of Navier-Stokes flow to ideal flow in Gevrey space.

Mathematical Methods in the Applied Sciences 40 (2017), 5161-5176 - Feng Cheng, Wei-Xi Li and
Chao-Jiang Xu

Gevery regularity with weight for incompressible Euler equation in the half plane.

Acta Mathematics Scientia, 37 (2017), no. 4, 1115-1132 - Wei-Xi Li

Compactness criteria for the resolvent of the Fokker-Planck operator.

Ann. Sc. Norm. Super. Pisa Cl. Sci. (doi: 10.2422/2036-2145.201511_008) - Wei-Xi Li, Peng Luo and Shuying Tian

$L^2$-regularity of kinetic equations with external potential.

Journal of Differential Equations 260 (2016), 5894-5911 - Wei-Xi Li, Di Wu and
Chao-Jiang Xu

Gevrey Class Smoothing Effect for the Prandtl Equation.

SIAM J. Math. Anal. 48 (2016),1672–1726 - Wei-Xi Li

Global hypoelliptic estimates for fractional order kinetic equation.

Mathematische Nachrichten 287(2014), 610-637 - Wei-Xi Li and Alberto
Parmeggiani

Gevrey-hypoellipticity for twisted Laplacians.

Journal of Pseudo-Differential Operators and Applications 4(2013), 279-296 - Hua
Chen, Wei-Xi Li and Ling-Jun Wang

Regularity of traveling free surface water waves with vorticity.

Journal of Nonlinear Science 23(2013), 1111-1142 - Frédéric
Hérau and Wei-Xi Li

Global hypoelliptic estimates for Landau-type operator with external potential.

Kyoto J. Math 53 (2013), 533-565 -
Renjun Duan and Wei-Xi Li

Hypocoercivity for the linear Boltzmann equation with confining forces.

Journal of Statistical Physics 148(2012), 306-324 - Wei-Xi Li

Global hypoellipticity and compactness of resolvent for Fokker-Planck operator.

Ann. Sc. Norm. Super. Pisa Cl. Sci. Vol. XI(2012), 789-815. - Hua
Chen, Wei-Xi Li and
Chao-Jiang Xu

Gevrey regularity of subelliptic Monge-Ampère equations in the plane.

Advances in Mathematics 228(2011) 1816-1841. - Hua
Chen, Wei-Xi Li and
Chao-Jiang Xu

Gevrey hypoellipticity for a class of kinetic equations.

Communications in Partial Differential Equations 36 (2011) 693-728. - Hua
Chen, Wei-Xi Li and
Chao-Jiang Xu

Analytic smoothness effect of solutions for spatially homogeneous Landau equation.

Journal of Differential Equations 248 (2010) 77-94. - Hua
Chen, Wei-Xi Li and
Chao-Jiang Xu

Gevrey hypoellipticity for linear and non-linear Fokker-Planck equations.

Journal of Differential Equations 246 (2009), 320- 339. - Hua
Chen, Wei-Xi Li and
Chao-Jiang Xu

Gevrey regularity for solution of the spatially homogeneous Landau equation.

Acta Mathematics Scientia 29(2009), 673-686. - Hua
Chen, Wei-Xi Li and
Chao-Jiang Xu

Propagation of Gevrey regularity for solutions of Landau equations.

Kinetic and Related Models 1(2008), 355- 368. - Shaohua Wu, Hua Chen and Wei-Xi Li

The local and global existence of the solutions of hyperbolic-parabolic system modeling biological phenomena.

Acta Mathematics Scientia 28 (2008), 101- 116.

## Accepted/published papers

## Collaborators & Mentors

Radjesvarane Alexandre Hua Chen Feng Cheng Nils Dencker Renjun Duan Frédéric Hérau Nicolas Lerner Peng Luo Van-Sang Ngo Alberto Parmeggiani Shuying Tian Ling-Jun Wang Xue Ping Wang Yan-Lin Wang Di Wu Shaohua Wu Chao-Jiang Xu Tong Yang