3D MHD Equations Research Articles

On the interior regularity conditions of a suitable weak solution to 3D MHD equations

On the interior regularity conditions of a suitable weak solution to 3D MHD equations

Stabilizing effect of the magnetic field and decay estimates for the 3D MHD equations with only one direction dissipation

This paper focuses on the stability and decay estimates of the 3D MHD equations with only one direction dissipation near a background magnetic field. The global stability and decay estimates of the 3D Navier-Stokes equations with only one direction dissipation remain an outstanding open problem. However, when coupled with the magnetic field in the MHD system involved here, the velocity field is shown to stabilize and decay in time. This result reveals that the magnetic field stabilizes and damps the electrically conducting fluids. By utilizing time-weighted methods and the method of bootstrapping argument, we establish that any perturbation near a magnetic field is globally stable in H3(R3). Furthermore, explicit decay rates in H2(R3) are obtained.

Upper Semicontinuity of Strong Pullback Attractors for Delay Non-autonomous Dynamical Systems and Application to Double Time-delayed 2D MHD Equations

Upper Semicontinuity of Strong Pullback Attractors for Delay Non-autonomous Dynamical Systems and Application to Double Time-delayed 2D MHD Equations

Global well-posedness and decay to 3D MHD equations with nonlinear damping

In this paper, we consider the Cauchy problem for the 3D MHD system with nonlinear damping terms κ|u|α−1u and γ|b|β−1b and prove the existence and uniqueness of global strong solution when one of the following two conditions is verified (1)3≤α<4,β≥6α−1+1;(2)α≥4,β≥1. This greatly improves the previous results. Moreover, we also discuss the uniqueness of weak solutions when α and β satisfy certain conditions. Finally, we study the decay of weak solutions when α,β>73 and obtain the upper bound for the derivative of strong solution when 3≤α<4,β≥6α−1+1 or α≥4,β>73.

On the Sobolev stability threshold for shear flows near Couette in 2D MHD equations

In this work, we study the Sobolev stability of shear flows near Couette in the 2D incompressible magnetohydrodynamics (MHD) equations with background magnetic field $(\alpha,0 )^\top$ on $\mathbb {T}\times \mathbb {R}$ . More precisely, for sufficiently large $\alpha$ , we show that when the initial datum of the shear flow satisfies $\left \| U(y)-y\right \|_{H^{N+6}}\ll 1$ , with $N&gt;1$ , and the initial perturbations ${u}_{\mathrm {in}}$ and ${b}_{\mathrm {in}}$ satisfy $\left \| ( {u}_{\mathrm {in}},{b}_{\mathrm {in}}) \right \| _{H^{N+1}}=\epsilon \ll \nu ^{\frac 56+\tilde \delta }$ for any fixed $\tilde \delta &gt;0$ , then the solution of the 2D MHD equations remains $\nu ^{-(\frac {1}{3}+\frac {\tilde \delta }{2})}\epsilon$ -close to $( e^{\nu t \partial _{yy}}U(y),0)^\top$ for all $t&gt;0$ .

Uniform attractors for 3D MHD equations with nonlinear damping

ABSTRACT The aim of this paper is to investigate the uniform attractors for 3D MHD equations with nonlinear damping. Some uniform estimates for strong solutions are established. Furthermore, we prove that 3D damped equations have an ( V , V ) -uniform attractor and an ( V , H 2 ) -uniform attractor, and the ( V , V ) -uniform attractor is actually the ( V , H 2 ) -uniform attractor.

Space–time decay rates for the 3D full compressible MHD equations

Space–time decay rates for the 3D full compressible MHD equations

Existence and Approximation of Statistical Solutions of the 3D MHD Equations

Existence and Approximation of Statistical Solutions of the 3D MHD Equations

Mean attractors and invariant measures of locally monotone and generally coercive SPDEs driven by superlinear noise

We study the global solvability, mean attractors and invariant measures for an abstract locally monotone and generally coercive SPDEs driven by infinite-dimensional superlinear noise defined in a dual space of intersection of finitely many Banach spaces. The main feature of this abstract system is that it covers a larger class of fundamental models which are included or not included previously. Under an extended locally monotone variational setting, we establish the global well-posedness, Itô's energy equality and existence of mean random attractors in some high-order Bochner spaces. The existence, uniqueness, support, (high-order and exponential) moment estimates, ergodicity, (pointwise and Wasserstein-type) exponentially mixing and asymptotic stability of invariant measures and evolution systems of measures are discussed for autonomous and nonautonomous stochastic equations. A stopping time technique is used to prove the convergence of solutions in probability in order to overcome the difficulty caused by the local monotonicity and superlinear growth of the coefficients. Our abstract results and unified methods are expected to be applied to various types of SPDEs like 2D Navier-Stokes equations, 2D MHD equations, 2D magnetic Bénard problem, Burgers type equations, 3D Leray α-model, convective Brinkman-Forchheimer equations, fractional (s,p)-Laplacian equations with monotone nonlinearities of polynomial growth of arbitrary order, and others.

Global regularity of 2D MHD equations with almost Laplacian velocity dissipation

We obtain the global existence and global regularity for the 2D MHD equations with almost Laplacian velocity dissipation, which require the dissipative operators weaker than any power of the fractional Laplacian. The result can be regarded as a further improvement and generalization of previous works.

The Global Well-posedness of Strong Solutions to 2D MHD Equations in Lei-Lin Space

The Global Well-posedness of Strong Solutions to 2D MHD Equations in Lei-Lin Space

Local Existence of Solutions to the Non-Resistive 3D MHD Equations with Power-law Type

Local Existence of Solutions to the Non-Resistive 3D MHD Equations with Power-law Type

Time Optimal Control Problem of the 2D MHD Equations with Memory

Time Optimal Control Problem of the 2D MHD Equations with Memory

Asymptotic stability of the 2D MHD equations without resistivity on a flat strip domain

In this paper, we establish the asymptotic stability of certain stationary solution to 2D magnetohydrodynamic flow without resistivity in the infinite flat strip Ω=R×(0,1). In addition, some explicit decay rates are also established.

Asymptotic stability of the 2D MHD equations without magnetic diffusion

We investigate the asymptotic stability of certain steady solution to the 2D MHD equations without magnetic diffusion in an infinite strip domain. Decay rates of smooth solution to that system are also given.

Global Small Solutions to a Special $$2\frac{1}{2}$$-D Compressible Viscous Non-resistive MHD System

This paper solves the global well-posedness and stability problem on a special \(2\frac{1}{2}\)-D compressible viscous non-resistive MHD system near a steady-state solution. The steady-state here consists of a positive constant density and a background magnetic field. The global solution is constructed in \(L^p\)-based homogeneous Besov spaces, which allow general and highly oscillating initial velocity. The well-posedness problem studied here is extremely challenging due to the lack of the magnetic diffusion and remains open for the corresponding 3D MHD equations. Our approach exploits the enhanced dissipation and stabilizing effect resulting from the background magnetic field, a phenomenon observed in physical experiments. In addition, we obtain the solution’s optimal decay rate when the initial data is further assumed to be in a Besov space of negative index.

Global Well-Posedness of the 2D MHD Equations of Damped Wave Type in Sobolev Space

The magnetohydrodynamic system of damped wave type (abbreviated as MHD-wave system) is formally derived from Maxwell's equations of electromagnetism by keeping the usually ignored small term involving the product of permittivity and magnetic permeability. When this term is ignored in the context of nonrelativistic charged fluid, one obtains the standard MHD system. This extra term in the MHD-wave system assumes the form $\gamma \partial_{tt} b$ with $\gamma>0$ being a small constant and $b$ the magnetic field. Mathematically this term makes the global well-posedness problem much more challenging than the corresponding MHD system. Even the global existence and regularity problem for the 2D MHD-wave system appears to be open. This paper solves the global well-posedness problem in a critical Sobolev setting when $\gamma$ and the size of the initial data satisfy a suitable constraint. In addition, the solution of the MHD-wave system is shown to converge to that of the corresponding MHD system with an explicit rate. The energy method does not work here, and this paper presents a new approach.

Regularization by transport noises for 3D MHD equations

We consider the problem of regularization by noise for the three dimensional magnetohydrodynamical (3D MHD) equations. It is shown that, in a suitable scaling limit, multiplicative noise of transport type gives rise to bounds on the vorticity fields of the fluid velocity and magnetic fields. As a result, if the noise intensity is big enough, then the stochastic 3D MHD equations admit a pathwise unique global solution for large initial data, with high probability.

Uniform attractors for nonautonomous 2D MHD equations with partial dissipation

In this article, we study the long-time behavior of solutions of nonautonomous 2D magnetohydrodynamics equations with partial dissipation and time-dependent external forcing, which is weakly normal. We prove that the system possesses strong uniform attractors. Furthermore, the structure of the uniform attractors is obtained, and the fractal dimension is estimated for the kernel sections of the uniform attractors.

Stability for the 2D MHD equations with horizontal dissipation

In this paper we consider the following 2D MHD system with horizontal dissipation in a strip domain T × R. ∂ t u + u · ∇ u + ∂ 11 u + ∇ p = b · ∇ b , ∂ t b + u · ∇ b + ∂ 11 b = b · ∇ u , ∇ · u = ∇ · b = 0 , u ( 0 ) = u 0 ( x ) , b ( 0 ) = b 0 ( x ) . A bootstrapping argument together with a more accurate energy functional is employed in order to get the stability for the above system. Moreover, using a suitable transform, we also investigate the 2D MHD system with vertical dissipation in a strip domain R × T.