THE GLOBAL EXISTENCE AND ANALYTICITY OF A MILD SOLUTION TO THE 3D REGULARIZED MHD EQUATIONS

In this paper,we study the three-dimensional regularized MHD equations with fractional Laplacians in the dissipative and diffusive terms.We establish the global existence of mild solutions to this system with small initial data.In addition,we also obtain the Gevrey class regularity and the temporal decay rate of the solution.

GLOBAL UNIQUE SOLUTIONS FOR THE INCOMPRESSIBLE MHD EQUATIONS WITH VARIABLE DENSITY AND ELECTRICAL CONDUCTIVITY

We study the global unique solutions to the 2-D inhomogeneous incompressible MHD equations,with the initial data(u0,B0)being located in the critical Besov space■and the initial densityρ0 being close to a positive constant.By using weighted global estimates,maximal regularity estimates in the Lorentz space for the Stokes system,and the Lagrangian approach,we show that the 2-D MHD equations have a unique global solution.

A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

In this paper,we construct a high-order discontinuous Galerkin(DG)method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics(VRMHD).To control the divergence error in the magnetic field,both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD.Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes,respectively,showing that the scheme can maintain the positivity-preserving(PP)property under some CFL conditions when combined with the strong-stability-preserving time discretization.Then,general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes.Numerical tests demonstrate the effectiveness of the proposed schemes.

ENERGY DISSIPATION FOR WEAK SOLUTIONS OF INCOMPRESSIBLE MHD EQUATIONS

In this article, we mainly study the local equation of energy for weak solutions of 3D MHD equations. We define a dissipation term D（u, B） that steins from an eventual lack of smoothness in the solution, and then obtain a local equation of energy for weak solutions of 3D MHD equations. Finally, we consider the 2D case at the end of this article.

WELL-POSEDNESS IN CRITICAL SPACES FOR THE FULL COMPRESSIBLE MHD EQUATIONS

In this paper we prove local well-posedness in critical Besov spaces for the full compressible MHD equations in R^N, N≥ 2, under the assumptions that the initialdensity is bounded away from zero. The proof relies on uniform estimates for a mixed hyperbolic/parabolic linear system with a convection term.

LOCAL WELL-POSEDNESS OF STRONG SOLUTIONS FOR THE NONHOMOGENEOUS MHD EQUATIONS WITH A SLIP BOUNDARY CONDITIONS

This article is concerned with the 3 D nonhomogeneous incompressible magnetohydrodynamics equations with a slip boundary conditions in bounded domain.We obtain weighted estimates of the velocity and magnetic field,and address the issue of local existence and uniqueness of strong solutions with the weaker initial data which contains vacuum states.

THE GLOBAL L^2 STABILITY OF SOLUTIONS TO THREE DIMENSIONAL MHD EQUATIONS

In this paper, we mainly study the global L2 stability for large solutions to the MHD equations in three-dimensional bounded or unbounded domains. Under suitable conditions of the large solutions, it is shown that the large solutions are stable. And we obtain the equivalent condition of this stability condition. Moreover, the global existence and the stability of two-dimensional MHD equations under three-dimensional perturbations are also established.

TIME DECAY RATE OF SOLUTIONS TO THE HYPERBOLIC MHD EQUATIONS IN R3

In this paper, we first show the global existence, uniqueness and regularity of weak solutions for the hyperbolic magnetohydrodynamics(MHD) equations in R^3. Then we establish that the solutions with initial data belonging to H^m(R^3) ∩ L^1(R^3) have the following time decay rate:║▽~mu(x, t) ║~2+║ ▽~mb(x, t)║~ 2+ ║▽^(m+1)u(x, t)║~ 2+ ║▽^(m+1)b(x, t) ║~2≤ c(1 + t)^(-3/2-m)for large t, where m = 0, 1.

GlobalWell-Posedness for the 2-DMHD Equations with Magnetic Diffusion

In this paper,we consider the 2-D MHD equations with magnetic resistivity but without dissipation on the torus.We prove that if the initial data is small in H4(T2),then the 2-D MHD equations are globally well-posed.To our knowledge,this is the first global well-posedness result for this system.

LOCAL REGULARITY CRITERIA OF A SUITABLE WEAK SOLUTION TO MHD EQUATIONS

We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitable weak solution are HSlder continuous near boundary provided that the scaled mixed Lx,t^p,q-norm of the velocity vector field with 3/p ＋ 2/q 〈 2, 2 〈 q 〈 ∞ is sufficiently small near the boundary. Also, we will investigate that for this 3 2〈3 solution U ∈ Lx,t^p,q with 1 〈 3＋p ＋2/q＋≤3/2, 3 〈 p 〈 ∞, the Hausdorff dimension of its singular set is no greater than max{p, q}（3/q＋2/q- 1）.

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

This paper focuses on the statistical characteristics of the 3D MHD equations.We firstly establish an existence theorem of a Vishik-Fursikov measure of the 3D MHD equations by taking advantage of the Krein-Milman theorem along with some functional and measure theories.Then by applying the Topsoe lemma on the constructed trajectory space possessing some special topological properties,we show that the Vishik-Fursikov measure and the stationary Vishik-Fursikov statistical solution of the 3D MHD system are approximated by the counterparts of the 3D MHD-αsystem,respectively,as the parameterαdecreases to zero.

On the Blow-up Criterion of Smooth Solutions to the 3D Ideal MHD Equations

In this paper, we consider the blow-up of smooth solutions to the 3D ideal MHD equations. Let (u, b) be a smooth solution in (0, T). It is proved that the solution (u, b) can be extended after t = T if . This is an improvement of the result given by Caflisch, Klapper, and Steele [3].

Global well-posedness of the MHD equations via the comparison principle

In this paper,we prove the global well-posedness of the incompressible magneto-hydrodynamics(MHD)equations near a homogeneous equilibrium in the domain R^k×T^(d-k),d≥2,k≥1 by using the comparison principle and constructing the comparison function.

Global well-posedness for the 3-D MHD equations with partial diffusion in the periodic domain

In this paper, we prove the global well-posedness of the 3-D magnetohydrodynamics(MHD) equations with partial diffusion in the periodic domain when the initial velocity is small and the initial magnetic field is close to a background magnetic field satisfying the Diophantine condition.

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 magnetohydrodynamic(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 value 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 equation as the dissipations tend to zero.

Energy Equality and Uniqueness of Weak Solutions to MHD Equations in L~∞(O,T;L^n(Ω))

In this paper, we study the energy equality and the uniqueness of weak solutions to the MHD equations in the critical space L∞（0, T; L^n（Ω）. We prove that if the velocity u belongs to the critical space L∞（0, T; L^n（Ω）, the energy equality holds. On the basis of the energy equality, we further prove that the weak solution to the MHD equations is unique.

A Note on the Blow-up Criterion of Smooth Solutions to the 3D Incompressible MHD Equations

In this note, we will give a new proof of the blow-up criterion of smooth solutions to the 3D incompressible magneto-hydrodynamic equations by a simple application of Gagliardo-Nirenberg＇ s inequality.

Well-Balanced Central Scheme for the System of MHD Equations with Gravitational Source Term

A well-balanced second order finite volume central scheme for the magnetohydrodynamic(MHD)equations with gravitational source term is developed in this paper.The scheme is an unstaggered central scheme that evolves the numerical solution on a single grid and avoids solving Riemann problems at the cell interfaces using ghost staggered cells.A subtraction technique is used on the conservative variables with the support of a known steady state in order to manifest the well-balanced property of the scheme.The divergence-free constraint of themagnetic field is satisfied after applying the constrained transport method(CTM)for unstaggered central schemes at the end of each time-step by correcting the components of the magnetic field.The robustness of the proposed scheme is verified on a list of numerical test cases from the literature.

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

In this paper, we study the Cauchy problem of the 2D incompressible magnetohydrodynamic equations in Lei-Lin space. The global well-posedness of a strong solution in the Lei-Lin space χ^(-1)(R^(2)) with any initial data in χ^(-1)(R^(2)) ∩ L^(2)(R^(2)) is established. Furthermore, the uniqueness of the strong solution in χ^(-1)(R^(2)) and the Leray-Hopf weak solution in L^(2)(R^(2)) is proved.

Global Well-posedness for the Non-viscous MHD Equations with Magnetic Diffusion in Critical Besov Spaces

In this paper,we mainly investigate the Cauchy problem of the non-viscous MHD equations with magnetic diffusion.We first establish the local well-posedness(existence,uniqueness and continuous dependence)with initial data(u_(0),b_(0))in critical Besov spaces B_(p,1)^(d/p+1)×B_(p,1)^(d/p)with 1≤p≤∞,and give a lifespan T of the solution which depends on the norm of the Littlewood–Paley decomposition(profile)of the initial data.Then,we prove the global existence in critical Besov spaces.In particular,the results of global existence also hold in Sobolev space C([0,∞);H~s(S~2))×(C([0,∞);H^(s-1)(S~2))∩L~2([0,∞);H~s(S~2)))with s>2,when the initial data satisfies∫_(S~2)b_(0)dx=0 and||u_(0)||B_(()∞,1~((S~2)))~1+||b_(0)||B_(()∞,1^(S~2))~0≤ε.It’s worth noting that our results imply some large and low regularity initial data for the global existence.