Global Existence of Strong Solutions to Incompressible MHD

We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for incompressible MHD equations in a bounded smooth domain of three spatial dimensions with initial density being allowed to have vacuum, in particular, the initial density can vanish in a set of positive Lebessgue measure. More precisely, under the assumption that the production of the quantities $|\sqrt\rho_0u_0|_{L^2(\Omega)}^2+|H_0|_{L^2(\Omega)}^2$ and $|\nabla u_0|_{L^2(\Omega)}^2+|\nabla H_0|_{L^2(\Omega)}^2$ is suitably small, with the smallness depending only on the bound of the initial density and the domain, we prove that there is a unique strong solution to the Dirichlet problem of the incompressible MHD system.


HUAJUN GONG AND JINKAI LI
Abstract. We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for the incompressible MHD equations in bounded smooth domains of R 3 under some suitable smallness conditions. The initial density is allowed to have vacuum, in particular, it can vanish in a set of positive Lebessgue measure. More precisely, under the assumption that the production of the quantities √ ρ 0 u 0 2 L 2 (Ω) + H 0 2 L 2 (Ω) and ∇u 0 2 L 2 (Ω) + ∇H 0 2 L 2 (Ω) is suitably small, with the smallness depending only on the bound of the initial density and the domain, we prove that there is a unique strong solution to the Dirichlet problem of the incompressible MHD system.

Introduction
Magnetohydrodynamics (MHD for short) is the study of the interaction between magnetic fields and moving conducting fluids, which can be described by the following system where ρ = ρ(x, t) ∈ R + denotes the density, u = u(x, t) ∈ R 3 the fluid velocity, H = H(x, t) ∈ R 3 the magnetic field, p = p(x, t) ∈ R the pressure, positive constant µ is called the kinematic viscosity and positive constant λ is called the magnetic diffusivity. Usually, we refer to equation ( There are a lot of literatures on the study of MHD. Global existence of weak solutions to the homogeneous incompressible MHD were proven in [3] and [4] long time ago, the density dependent case was later proven in [5] by using Lions's method [6]. The corresponding results on the global existence of weak solutions to the compressible MHD are proven in [8][9][10]13] by using the method exploited in [7] (see also [11,12]), where in [8] the isentropic case is considered, while in [9,10,13] the nonisothermal model is considered. Local existence and uniqueness of strong solutions can be found in [14] and [15] for the incompressible model. In [17] and [18] for the compressible model, where in the last three papers the non-isothermal model are considered. Global existence and uniqueness of strong solutions to the incompressible MHD with vacuum for two dimensional case are proved in [16], recently, and this work is new even for the inhomogeneous incompressible Navier-Stokes equations, however the three dimensional case is proven in [19] with small initial data and away from vacuum. Later the regularity criteria for the solution to the 3D MHD are proved in [21][22][23].
One of the most important question for system (1.1)-(1.4) is to prove the global existence and uniqueness of solutions (ρ, u, H) satisfying the initial condition (ρ, u, H)| t=0 = (ρ 0 , u 0 , H 0 ) in Ω, (1.5) and boundary conditions where n is the unit outward norm vector on ∂Ω. The aim of this paper is to prove the global existence and uniqueness of strong solutions to system (1.1)-(1.6) with the initial data being allowed to have vacuum. For 1 ≤ p ≤ ∞, we denote by u p the L p (Ω) norm of the function u. The definition of strong solution is stated in the following: in Ω × (0, T ) for some pressure function p, and satisfies the initial condition (1.5) and boundary condition (1.6), with the regularity Our main result is stated bellow: Let Ω be a bounded smooth domain in R 3 andρ a positive number. Assume that the initial data (ρ 0 , u 0 ) satisfies the conditions and the compatibility condition Then there is a positive constant ε 0 depending only onρ and Ω, such that, if then the system (1.1)-(1.6) has a unique global strong solution.
and thus Theorem 1.1 can be viewed as a result on the global existence of strong solutions with vacuum in critical space. It seems that this is the first result in this direction, even for the Navier-Stokes equations.

Proof of Theorem 1.1
Throughout this section, we denote √ ρ 0 u 0 where the function Φ(T ) is given by We will use the following lemma, which states the local existence and blow up criterion of the local strong solutions. To proof the global existence of strong solutions, we need to extend the local strong solution given in the above lemma to be a global one. For this purpose, we need do some a priori estimates on the local strong solutions. The following two lemmas give the energy estimates on the strong solutions, where the first one concerns the basic energy estimates, while the second one concerns the higher order estimates. Let (ρ, u, H) be a strong solution to system (1.1)-(1.6) on (0, T ). Then, there holds for any t ∈ (0, T ).
Proof. Multiply (1.2) by u and integrate over Ω, by the aid of (1.1), we obtain after integration by parts that for any t ∈ (0, T ).
By using the two lemmas above, we can prove the following a priori estimates.
In view of the regularities of u and H, one can easily check that both E(t) and Φ(t) are continuous functions on [0, T ]. By Lemma 2.3, there is a positive constant C * , such that (2.9) We take and suppose that Otherwise, by the continuity and monotonicity of Φ(t), there is T 0 ∈ (0, T ], such that (2.10) On account of (2.10), it follows from (2.9) that and thus E(T 0 ) ≤ 4( ∇u 0 2 2 + ∇H 0 2 2 ). Recalling the definition of E(t) and Φ(t), we deduce from the above inequality that contradicting to (2.10). This contradiction implies that the claim is true.
By the aid of the claim we proved in the above, it follows from (2. 2 ) ≤ ε 0 . By Lemma 2.1, there is a unique strong solution (ρ, u, H) to system (1.1)-(1.6). Extend such local solution to the maximal existence time interval [0, T * ). We will prove that T * = ∞. Suppose, by contradiction, that T * < ∞. By Lemma 2.1, the time T * can be characterized as follows contradicting to (2.11). This contradiction provides us that T * = ∞, and thus we obtain a global strong solution. The proof is complete.