Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density

In this paper, we consider the Cauchy problem of the incompressible MHD system with discontinuous initial density in ${\mathbb R}^3$. We establish the global well-posedness of the MHD system if the initial data satisfies \begin{document}$(ρ_0, u_0, H_0)∈ L^{∞}({\mathbb R}^3)× H^s({\mathbb R}^3)× H^s({\mathbb R}^3)$\end{document} with \begin{document}$\frac{1}{2} and \begin{document}$0 for some small \begin{document}$c>0$\end{document} which only depends on \begin{document}$\underline{ρ}, \overline{ρ}$\end{document} . As a byproduct, we also get the decay estimate of the solution.


(Communicated by Zhouping Xin)
Abstract. In this paper, we consider the Cauchy problem of the incompressible MHD system with discontinuous initial density in R 3 . We establish the global well-posedness of the MHD system if the initial data satisfies (ρ 0 , u 0 , H 0 ) ∈ L ∞ (R 3 ) × H s (R 3 ) × H s (R 3 ) with 1 2 < s ≤ 1 and 0 < ρ ≤ ρ 0 ≤ ρ < +∞, (u 0 , H 0 ) 1. Introduction and the main results. Magnetohydrodynamic (MHD) system describes the interaction between the magnetic field and conductive fluid, which is a nonlinear system that couples Navier-Stokes equations with Maxwell equations. In magnetohydrodynamics, the displacement currents can be neglected in the time dependent Maxwell equations. So it becomes the following system: 2010 Mathematics Subject Classification. Primary: 35Q35, 76D03; Secondary: 35B40. Key words and phrases. Global well-posedness, incompressible MHD system, Cauchy problem, decay estimate.
The first author is supported by the Postdoctoral Science Foundation of China grant 2017M620688, the second author is supported by NSFC grant 11731014,11571254 and the third author is supported by NSFC grant 11601533. * Corresponding author: Xiaoping Zhai. where ρ is the density, u is the velocity field, H is the magnetic field and P is the scalar pressure. The body force curlH × H = H · ∇H − ∇ |H| 2 2 and curl(u × H) = H · ∇u − u · ∇H if divu = divH = 0.
When the density ρ is not a constant, (1) is the so called inhomogeneous incompressible MHD system. Compared with the Navier-Stokes equations, the dynamic motion of the fluid and the magnetic field interact on each other and both the hydrodynamic and electrodynamic effects in the motion are strongly coupled, the problems of MHD system are considerably more complicated. Even though, in the past several years, there are also many mathematical results related to the incompressible MHD system (see [3], [4], [8], [11], [15], [16], [18], [26], [27], [34], [36], [37], [38]). Gerbeau and Le Bris [18] and also Desjardins and Le Bris [15] researched global existence of weak solution of finite energy in R 3 and in the torus T 3 . Chen, Tan and Wang [11] showed the local strong solution in H 2 when the initial data contain vacuum states (i.e. the initial density ρ may vanish in some open set of Ω). Then, Huang and Wang [26] extended the local strong solution to be global in two dimensions ( see also Gong and Li [19] for three dimensions).
When the initial density is away from zero and is close enough to a positive constant, Abidi and Hmidi in [3] and Abidi and Paicu in [4] obtained the global solution with small initial data in the critical Besov spaces. By critical, we mean that we want to solve the system (1) in functional spaces with invariant norms by the changes of scales which leaves (1) invariant. In the case of inhomogeneous incompressible MHD fluids, it is easy to see that the transformations: (ρ λ , u λ , H λ )(t, x) = (ρ(λ 2 t, λx), λu(λ 2 t, λx), λH(λ 2 t, λx)) have that property, provided that the pressure term has been changed accordingly. The results in [3], [4] have been extended by Chen, Li and Xu in [8], Zhai, Li and Xu in [36], Zhai, Li and Yan in [37].
When the initial density is away from zero and is not close enough to a positive constant, Gui in [20] considered the global well-posedness of 2-D MHD equations with constant viscosity and variable conductivity for a generic family of the variations of the initial data, and established the global well-posedness of the equations in the critical spaces with constant conductivity. Zhai and Yin in [38] got the global solution in R 3 by applying a new a priori estimate for an elliptic equation with nonconstant coefficients. Hoff [23,24] and Huang, Li and Xin [25] respectively proved the global existence of small energy weak solutions and global well-posedness of small energy classical solutions of the isentropic compressible Navier-Stokes equations, where, [25] is the first for global classical solutions that may have large oscillations and can contain vacuum states. An important idea in [23], [24] and [25] is that by using an appropriate time weight, the energy estimate for space derivatives of the velocity field can be closed although the initial date has low regularity (even only in L 2 (R d )). By using a similar idea, Chen, Zhang and Zhao in [9] obtained the Fujita-Kato solution for the 3-D inhomogeneous Navier-Stokes equations.
In this paper, we will get a similar result to [9] for the 3-D inhomogeneous incompressible MHD system. We establish the global existence and uniqueness of the solution, under the condition that the initial date (u 0 , H 0 ) is small in the critical spaceḢ 1 2 . We use the idea of time weight in [9], [23], [24] and [25] to deal with less regular initial velocity field and magnetic field, and the Lagrangian idea in [13], [14] to deal with the proof of the uniqueness of the solution. At last, we also get the decay estimate of the solution by the dual method as a generalization of the result in [9].
The main results of this paper are the following theorems: and there exists a constant c depending only on ρ, ρ such that then the system (1) has a unique global solution (ρ, u, H) satisfying where the constant C depending only on ρ, ρ and in which ω(t) = min{1, t}.

FEI CHEN, BOLING GUO AND XIAOPING ZHAI
Which means that for u 0 , H 0 ∈ H s (R 3 ) with s ∈ (0, 1), (5) and The aim of the introduction of the time weight ω(t) in Theorem 1.1 is to close the energy estimates for space derivatives of the velocity field and the magnetic field with low regular initial data Here we only consider the low regular case when 1 2 < s ≤ 1, since it is much easier to deal with the higher regular case.
We also get the following decay estimate of the solution obtained in Theorem 1.1: 2]. Then the solution satisfies the following inequality . Throughout this paper, we use the following notations. For simplicity, we denote For 1 ≤ p ≤ ∞, and m ∈ N, the Sobolev spaces are defined in a standard manner: Given a Banach space X, we shall denote by (a, b) X = a X + b X . The rest of the paper unfolds as follows. In Section 2, we prove the global existence part of Theorem 1.1 by using the time weighted energy method; in Section 3, we complete the uniqueness of the solutions by using the Lagrangian coordinate method; in Section 4, we further prove the decay of the solution by applying the dual method.
2. The proof of the existence part of Theorem 1.1. In this section, we concentrate on the proof of the existence part of Theorem 1.1. Let j σ be the standard Friedrich's mollifier, and define In what follows, we'll only present the uniform estimates (4) of the smooth approximate solutions (ρ σ , u σ , H σ ). Then, the existence part of Theorem 1.1 essentially follows by (4) and a standard compactness argument. In the sequel, we omit the superscript σ for simplicity. And we denote by C or C i (i = 1, 2) the positive constant, which may depend on ρ, ρ but does not rely on the time T and the superscript σ.
Next, we'll use the bootstrap theory to get the uniform estimate. We suppose that the following a priori hypothesis holds: for some small enough c 0 > 0 determined later. Next, we'll get the estimate (4) and deduce the following refined estimate Step 1. L 2 and H 1 estimates without the time weight.
From the transport equation, we can easily get and with (2), we have Taking L 2 inner product of (1) 2 with u, integrating by parts over R 3 , and using the transport equation (1) Similarly, taking L 2 inner product of (1) 3 with H, we obtain By combining (8) with (9) and integrating it over [0, t], ∀ t ∈ [0, T ], we have Taking L 2 inner product of (1) 2 with u t and integrating by parts over R 3 , we obtain Similarly, taking L 2 inner product of (1) 3 with H t , we achieve We rewrite (1) 2 as which along with the classical estimates of the Stokes equation leads to Similarly, By combining (13) with (14), we obtain and thus Summing up (11) and (12), we get by (15) that which shows that provided the constant c 0 in (6) satisfies For any t ∈ [0, T ], integrating (16) on [0, t], we can get by (15) that,

GLOBAL SOLUTION TO THE 3-D MHD SYSTEM 43
Step 2. H 1 estimate with the time weight and interpolation.
Noting that (u, H) is under the assumption (6), and by using an interpolation argument, we will get the estimate of F 1 (t). Considering the following linear system where P includes the magnetic pressure. From the proof of (17) and (18), we get Similar to the analysis in [32], by the complex interpolation, we deduce that for any θ ∈ [0, 1], By choosing c ≤ c0 2C in (3), we get (7). Then we complete the bootstrap arguments.
Differentiating (1) 2 with respect to t, taking the L 2 inner product with u t , and integrating by parts over R 3 , we get by (1) 1 that By similar arguments as (21), from (1) 3 , we obtain By combining (21) with (22), we have Define Multiplying ω(τ ) 2−s on both sides of (23), integrating with respect to τ and then integrating by parts over R 3 , we get From (19), Hölder's inequality, Sobolev inequality in [17], and by choosing c 0 ≤ 1 16C , we get that In addition, we deduce by Gagliardo-Nirenberg inequality and Young's inequality that and Similarly, we get that and

FEI CHEN, BOLING GUO AND XIAOPING ZHAI
Then, by substituting (25)- (30) into the summation of (24), we have So, by Gronwall's inequality, we obtain We get by interpolation, (19) and (20) that and if t ≥ 1, then so, by substituting (32) and (33) into the summation of (31) and (15), we get Since we get the uniform energy estimates, the proof of the existence part is completed.
3. The proof of the uniqueness part of Theorem 1.1.

3.1.
More regularity of the solutions. In this section, the aim is to prove the uniqueness of the solutions in Theorem 1.1. Firstly, we'll focus on some more information on the regularity of the solutions, which will be used in the proof.  If (ρ, u, H) is the solution of system (1) obtained in Theorem 1.1, T ∈ [0, 1] and γ ∈ [0, 1], then Similarly, we get that and Since−∆u + ∇P = −ρ∂ t u − ρu · ∇u + curlH × H, by the W 2,p estimate of the Stokes system and noting that 1 − s + 2γ(2−s) Then by Gagliardo-Nirenberg inequality and (4), we obtain that Similarly, we get that .
Then the proof of Lemma 3.1 is completed.
3.2. The Lagrangian formulation. Next, we show the Lagrangian formulation which will play a vital role in the proof. Define the trajectory of X(t, y) of u(t, x) by ∂ t X(t, y) = u(t, X(t, y)), X(0, y) = y, which shows the following relation between the Eulerian coordinate x and the Lagrangian coordinate y: By choosing T small enough, we can get from Lemma 3.1 that which ensures that for t ≤ T , X(t, y) is invertible with respect to the variable y, and we denote its inverse mapping Y (t, x).
Define v(t, y) = u(t, x) = u(t, X(t, y)), A(t, y) = (D y X(t, y) Here and in the sequel, A t denotes the transpose matrix of A, and A t : ∇ y means T r(A t ∇ y ). We denote as in [32] that ∇ u = A t · ∇ y , div u = div(A·), ∆ u = div u ∇ u , η(t, y) = ρ(t, X(t, y)), v(t, y) = u(t, X(t, y)), B(t, y) = H(t, X(t, y)), and Π(t, y) = P (t, X(t, y)). (40) So the Lagrangian formulation of (1) becomes Then, we transform the regularity information of the solution in the Eulerian coordinates into that in the Lagrangian coordinates.  and if γ < s, then we have Here the constant C depends on (u 0 , H 0 ) H s .

FEI CHEN, BOLING GUO AND XIAOPING ZHAI
Finally, we estimate ∇A . Thanks to (38), for any t ≤ T , we have By choosing T in (38) small enough, noting |∇u| = |Du|, we get from (42), (43) and Lemma 3.1 that for any t ≤ T and γ < s, This complete the proof of Lemma 3.2.
be a time independent positive function, and be bounded away from zero. Let R satisfy R t ∈ L 2 ((0, T )×R d ) and ∇divR ∈ L 2 ((0, T )× R d ). Then the following system has a unique solution (v, ∇Π) such that where the constant C depends on inf η and sup η, but is independent of T .
3.4. The proof of Lemma 3.4. In the following, we choose t small enough so that Denote ε(t) a function of t tending to zero as t → 0, which may be different in different lines.
Step 1. Estimate of δf 1 L 2 t (L 2 ) . By Lemma 3.2, we have We get by (44) that