arXiv subjects

Mengni Li

Publications and source records attributed to Mengni Li.

8 recordsLinked to original sources

arXiv2025

Existence and nonexistence of viscosity solutions for a class of degenerate/singular eigenvalue type equations

This paper is devoted to a complete classification on the existence and nonexistence results of viscosity solutions to the general Dirichlet problem for a class of eigenvalue type equations. With the distance function included in the right-hand side, this type of equations can be degenerate and (or) singular near the boundary of uniformly convex domains. One highlight is that all cases related to the exponent of the distance function are investigated. Moreover, when viscosity solutions exist, we derive a series of global estimates based on the distance function. The key ingredients of this paper include adaptions of the Perron method and comparison principle as well as constructions of suitable classical sub-solutions and super-solutions.

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arXiv2025

Pointwise boundary estimates for fully nonlinear elliptic equations with nonzero Dirichlet boundary conditions

In this paper, we investigate boundary estimates for the Dirichlet problem for a class of fully nonlinear elliptic equations with general boundary conditions, including nonzero boundary conditions. Given specific structural conditions on the problem, we develop pointwise boundary upper and lower bound estimates for convex solutions based on the subsolution and supersolution method. The global H\"older regularity can be derived as a direct consequence of these pointwise boundary estimates. These results fundamentally hinge on careful descriptions of the convexity properties of both the domains and the functions involved. Moreover, previous results on Monge-Amp\`ere equations with nonzero Dirichlet boundary conditions can be regarded as a special case of our results.

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arXiv2025

Ideal MHD. Part II: Rigidity from infinity for ideal Alfv\'en waves in 3D thin domains

This paper concerns the rigidity from infinity for Alfv\'en waves governed by ideal incompressible magnetohydrodynamic equations subjected to strong background magnetic fields along the $x_1$-axis in 3D thin domains $\Omega_\delta=\mathbb{R}^2\times(-\delta,\delta)$ with $\delta\in(0,1]$ and slip boundary conditions. We show that in any thin domain $\Omega_\delta$, Alfv\'en waves must vanish identically if their scattering fields vanish at infinities. As an application, the rigidity of Alfv\'en waves in $\Omega_{\delta}$, propagating along the horizontal direction, can be approximated by the rigidity of Alfv\'en waves in $\mathbb{R}^2$ when $\delta$ is sufficiently small. Our proof relies on the uniform (with respect to $\delta$) weighted energy estimates with a position parameter in weights to track the center of Alfv\'en waves. The key issues in the analysis include dealing with the nonlinear nature of Alfv\'en waves and the geometry of thin domains.

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arXiv2024

Zero-viscosity Limit for Boussinesq Equations with Vertical Viscosity and Navier Boundary in the Half Plane

In this paper we study the zero-viscosity limit of $2$-D Boussinesq equations with vertical viscosity and zero diffusivity, which is a nonlinear system with partial dissipation arising in atmospheric sciences and oceanic circulation. The domain is taken as $\mathbb{R}_+^2$ with Navier-type boundary. We prove the nonlinear stability of the approximate solution constructed by boundary layer expansion in conormal Sobolev space. The expansion order and convergence rates for the inviscid limit are also identified in this paper. Our paper extends a partial zero-dissipation limit results of Boussinesq system with full dissipation by Chae D. $[Adv. Math. 203, no. 2, 2006]$ in the whole space to the case with partial dissipation and Navier boundary in the half plane.

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arXiv2023

Existence, uniqueness and interior regularity of viscosity solutions for a class of Monge-Ampère type equations

The Monge-Ampère type equations over bounded convex domains arise in a host of geometric applications. In this paper, we focus on the Dirichlet problem for a class of Monge-Ampère type equations, which can be degenerate or singular near the boundary of convex domains. Viscosity subsolutions and viscosity supersolutions to the problem can be constructed via comparison principle. Finally, we demonstrate the existence, uniqueness and interior regularity (including $W^{2,p}$ with $p\in(1,+\infty)$, $C^{1,μ}$ with $μ\in(0,1)$, and $C^\infty$) of the viscosity solution to the problem.

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arXiv2022

Ideal MHD. Part III: Inverse scattering of Alfvén waves in three dimensional ideal magnetohydrodynamics

The purpose of this paper is to solve the inverse scattering problem of nonlinear Alfvén waves governed by the three dimensional ideal incompressible MHD system. Bridging together geometric methods and weighted energy estimates, we establish a couple of scattering isomorphisms to substantially strengthen our previous rigidity results. This answer is consistent with the physical intuition that Alfvén waves behave exactly in the same manner as their scattering fields detected by the faraway observers. The novelty of the present work is twofold: for one thing, the relationship between Alfvén waves emanating from the plasma and their scattering fields at infinities is explored to the best; for another thing, the null structure inherent in MHD equations is thoroughly examined, especially when we estimate the pressure term.

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arXiv2022

Entire subsolutions of a kind of k-Hessian type equations with gradient terms

In this paper, we consider a kind of $k$-Hessian type equations $S_k^{\frac{1}{k}}(D^2u+μ|D u|I)= f(u)$ in $\mathbb{R}^n$, and provide a necessary and sufficient condition of $f$ on the existence and nonexistence of entire admissible subsolutions, which can be regarded as a generalized Keller-Osserman condition. The existence and nonexistence results are proved in different ranges of the parameter $μ$ respectively, which embrace the standard Hessian equation case ($μ=0$) by Ji and Bao (Proc Amer Math Soc 138: 175--188, 2010) as a typical example. The difference between the semilinear case ($k=1$) and the fully nonlinear case ($k\ge 2$) is also concerned.

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arXiv2020

On the rigidity from infinity for nonlinear Alfvén waves

The Alfvén waves are fundamental wave phenomena in magnetized plasmas and the dynamics of Alfvén waves are governed by a system of nonlinear partial differential equations called the MHD system. In this paper, we study the rigidity aspect of the scattering problem for the MHD equations: We prove that the Alfvén waves must vanish if their scattering fields vanish at infinities. The proof is based on a careful study of the null structure and a family of weighted energy estimates.

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