arXiv subjects

Yuelong Xiao

Publications and source records attributed to Yuelong Xiao.

6 recordsLinked to original sources

arXiv2024

On the stability and exponential decay of the 3D MHD system with mixed partial dissipation near a equilibrium state

A main result of this paper establishes the global stability of the 3D MHD equations with mixed partial dissipation near a background magnetic field in the domain $Ω=\mathbb{T}^2\times\mathbb{R}$ with $\mathbb{T}^2=[0, 1]^2$. More precisely, each velocity equation lacks its own directional dissipation, and the magnetic equation lacks vertical dissipation in the MHD system. The key point to obtain the stability result is that we decompose the solution $(u,b)$ into the zeroth horizontal mode and the non-zeroth modes and complete the desired bound with the strong Poincaré type inequalities in the treatment of several nonlinear terms. Then we focus on the large-time behavior of the solution, where the non-zeroth modes decay exponentially in $H^2$, and the solution converges to its zeroth horizontal mode.

math.AP↗

arXiv2023

Stability and inviscid limit of the 3D anisotropic MHD system near a background magnetic field with mixed fractional partial dissipation

A main result of this paper establishes the global stability of the three-dimensional MHD equations near a background magnetic field with mixed fractional partial dissipation with $α, β\in(\frac{1}{2}, 1]$. Namely, the velocity equations involve dissipation $(Λ_1^{2α} + Λ_2^{2α}+σΛ_3^{2α})u$ with the case $σ=1$ and $σ=0$. The magnetic equations without partial magnetic diffusion $Λ_i^{2β} b_i $ but with the diffusion $(-Δ)^βb$, where $Λ_i^{s} (s>0)$ with $i=1, 2, 3$ are the directional fractional operators. Then we focus on the vanishing vertical kinematic viscosity coefficient limit of the MHD system with the case $σ=1$ to the case $σ=0$. The convergent result is obtained in the sense of $H^1$-norm.

math.AP↗

arXiv2020

On the vanishing dissipation limit for the incompressible MHD equations on bounded domains

In this paper, we investigate the solvability, regularity and the vanishing dissipation limit of solutions to the three-dimensional viscous magneto-hydrodynamic (MHD) equations in bounded domains. On the boundary, the velocity field fulfills a Navier-slip condition, while the magnetic field satisfies the insulating condition. It is shown that the initial-boundary problem has a global weak solution for a general smooth domain. More importantly, for a flat domain, we establish the uniform local well-posedness of the strong solution with higher order uniform regularity and the asymptotic convergence with a rate to the solution of the ideal MHD as the dissipation tends to zero.

math.AP↗

arXiv2017

Vanishing Viscosity Limit For the 3D Nonhomogeneous Incompressible Navier-Stokes Equations With a Slip Boundary Condition

In this paper, we investigate the vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier-Stokes equations with a slip boundary condition. We establish the local well-posedness of the strong solutions for initial boundary value problems for such systems. Furthermore, the vanishing viscosity limit process is established and a strong rate of convergence is obtained as the boundary of the domain is flat. In addition, it is needed to add some additional condition for density to match well the boundary condition.

math.AP↗

arXiv2013

On the Inviscid Limit of the 3D Navier-Stokes Equations with Generalized Navier-slip Boundary Conditions

In this paper, we investigate the vanishing viscosity limit problem for the 3-dimensional (3D) incompressible Navier-Stokes equations in a general bounded smooth domain of $R^3$ with the generalized Navier-slip boundary conditions (\ref{VSg}). Some uniform estimates on rates of convergence in $C([0,T],L^2(Ω))$ and $C([0,T],H^1(Ω))$ of the solutions to the corresponding solutions of the idea Euler equations with the standard slip boundary condition are obtained.

math.AP↗

arXiv2012

On 3D Lagrangian Navier-Stokes $α$ model with a Class of Vorticity-Slip Boundary conditions

This paper concerns the 3-dimensional Lagrangian Navier-Stokes $α$ model and the limiting Navier-Stokes system on smooth bounded domains with a class of vorticity-slip boundary conditions and the Navier-slip boundary conditions. It establishes the spectrum properties and regularity estimates of the associated Stokes operators, the local well-posedness of the strong solution and global existence of weak solutions for initial boundary value problems for such systems. Furthermore, the vanishing $α$ limit to a weak solution of the corresponding initial-boundary value problem of the Navier-Stokes system is proved and a rate of convergence is shown for the strong solution.

math.AP↗