arXiv subjects

Lianyun Peng

Publications and source records attributed to Lianyun Peng.

6 recordsLinked to original sources

arXiv2026

Nonlinear stability of a background magnetic field for the 3D compressible MHD equations with anisotropic dissipation

We study the nonlinear stability of an equilibrium with a background magnetic field for the three-dimensional compressible magnetohydrodynamic (MHD) equations in the whole space $\mathbb{R}^3$, in the strongly anisotropic regime where the velocity is dissipated only in the horizontal directions and the magnetic field is diffused in a single direction. We prove that for initial data sufficiently close to the equilibrium in a Sobolev space, the system admits a unique global-in-time solution that remains close to the equilibrium and enjoys quantitative dissipation estimates. The proof overcomes the severe lack of dissipation through two mechanisms: the background magnetic field is shown to generate enhanced dissipation for the magnetic field and the density, while a nonlinear cancellation mechanism is devised to resolve the loss of vertical derivatives caused by the compressible coupling.

math.AP↗

arXiv2026

Global uniform regularity and vanishing vertical viscosity limit for the compressible Navier--Stokes equations in the half-space

In geophysical flows such as large-scale ocean dynamics, the vertical viscosity is often much smaller than the horizontal viscosity. This anisotropy makes it natural to ask whether solutions of the full anisotropic compressible Navier--Stokes equations converge, as the vertical viscosity coefficient $\varepsilon \to 0$, to solutions of a horizontally dissipative limit system, and whether this limit can be justified globally in time. Prior work has answered this question locally in time or in the incompressible setting. We resolve this problem for the three-dimensional compressible Navier--Stokes equations in the upper half-space with the Navier slip boundary condition. This paper establishes two main results for small perturbations of a constant equilibrium state. First, we prove the existence of a unique global-in-time solution whose conormal Sobolev norm remains uniformly bounded for all $t \ge 0$ and all $\varepsilon \in (0,1)$. Second, we justify the global vanishing vertical viscosity limit. More precisely, we show that the solutions converge to a global solution of the horizontally dissipative compressible Navier--Stokes system. This provides the first rigorous justification of the anisotropic viscosity limit for compressible flows on an infinite time interval.

math.AP↗

arXiv2026

Global uniform regularity for the 3D compressible MHD equations near a background magnetic field

This paper resolves the global regularity problem for the three-dimensional compressible magnetohydrodynamics (MHD) equations in the three-dimensional whole space, in the presence of a background magnetic field. Motivated by geophysical applications, we consider an anisotropic compressible MHD system with weak dissipation in the $x_2$ and $x_3$ directions and small vertical magnetic diffusion. By exploiting the stabilizing effect induced by the background magnetic field and constructing a hierarchy of four energy functionals, we establish global-in-time uniform bounds that are independent of the viscosity in the $x_2$ and $x_3$ directions and the vertical resistivity. A key innovation in our analysis is the development of a two-tier energy method, which couples the boundedness of vertical derivatives with the decay of horizontal derivatives. The analysis of time scale, together with global regularity estimates and sharp decay rates, enable us to rigorously justify the vanishing dissipation limit and derive explicit long-time convergence rates to the compressible MHD system with vanishing dissipation in the $x_2$ and $x_3$ directions and no vertical magnetic diffusion. In the absence of magnetic field and background magnetic field, the global-in-time well-posedness and vanishing viscosity limit for the 3D compressible Navier-Stokes equations with only one direction dissipation remains a challenging open problem. This work reveals the mechanism by which the magnetic field enhances dissipation and stabilizes the fluid dynamics in the global well-posedness and vanishing viscosity limit.

math.AP↗

arXiv2025

Global uniform regularity for the 3D incompressible MHD equations with slip boundary condition near a background magnetic field

This paper resolves the global regularity problem for the three-dimensional incompressible magnetohydrodynamics (MHD) equations in the upper half-space with slip boundary conditions, in the presence of a background magnetic field. Motivated by geophysical applications, we consider an anisotropic MHD system with weak dissipation in the $x_2$ and $x_3$ directions and small vertical magnetic diffusion. By exploiting the stabilizing effect induced by the background magnetic field and constructing a hierarchy of four energy functionals, we establish global-in-time uniform bounds that are independent of the viscosity in the $x_2$ and $x_3$ directions and the vertical resistivity. A key innovation in our analysis is the development of a two-tier energy method, which couples the boundedness of conormal derivatives with the decay of tangential derivatives. These global conormal regularity estimates, together with sharp decay rates, enable us to rigorously justify the vanishing dissipation limit and derive explicit long-time convergence rates to the MHD system with vanishing dissipation in the $x_2$ and $x_3$ directions and no vertical magnetic diffusion. In the absence of a magnetic field, the global-in-time vanishing viscosity limit for the 3D incompressible Navier-Stokes equations with anisotropic dissipation remains a challenging open problem. This work reveals the mechanism by which the magnetic field enhances dissipation and stabilizes the fluid dynamics in the vanishing viscosity limit.

math.AP↗

arXiv2024

Local existence and uniqueness of solution to the two-dimensional inhomogeneous Prandtl equations by energy method

In this paper, we consider the local existence and uniqueness result for the inhomogeneous Prandtl equations in dimension two by energy method. First of all, for the homogeneous case, the local-in-time well-posedness theory of unsteady Prandtl equations was obtained by [Alexandre, Wang, Xu, Yang, J. Am. Math. Soc., 28 (3), 745-784 (2015)] and [Masmoudi, Wong, Comm. Pure Appl. Math., 68 (10), 1683-1741 (2015)] independently by energy method without any transformation. However, for the inhomogeneous case, the appearance of density will create some new difficulties for us to overcome the loss of tangential derivative of horizontal velocity. Thus, our first result is to overcome the loss of tangential derivative such that one can establish the local-in-time well-posedness result for the inhomogeneous Prandtl equations by energy method. Secondly, for the homogeneous case, the local-in-x well-posedness in higher regular space for the steady Prandtl equations was obtained by [Guo, Iyer, Comm. Math. Phys., 382 (3), 1403-447 (2021)] by energy method since they firstly found the good quantity(called `quotient'). With the help of this quotient, our second result is to establish the local-in-x well-posedness in higher regular Sobolev space for the steady inhomogeneous Prandtl equations.

math.AP↗

arXiv2023

Well-posedness and exponential stability of the inhomogeneous anisotropic incompressible Navier-Stokes equation with far-field vacuum in two-dimensional whole space

In this paper, we investigate the well-posedness theory and exponential stability for the inhomogeneous incompressible Navier-Stokes equation with only horizontal dissipative structure. Due to the lack of the vertical dissipative term and appearance of vacuum, it is a highly challenging tricky problem for us to study the well-posedness, stability and large-time behavior problems in two-dimensional whole space. The local-in-time well-posedness theory is successfully established at first because we develop some good estimates for the density and vorticity to control the nonlinear term. Finally, these good estimates of density and vorticity help us to establish the global-in-time well-posedness and exponential stability if the initial velocity is suitable small.

math.AP↗