On existence of global classical solutions to the 3D compressible MHD equations with vacuum

In this paper, the existence of global classical solutions is justified for the three-dimensional compressible magnetohydrodynamic (MHD) equations with vacuum. The main goal of this paper is to obtain a unique global classical solution on R3×[0,T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{3}\times [0, T]$\end{document} with any T∈(0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T\in (0, \infty )$\end{document}, provided that the initial magnetic field in the L3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{3}$\end{document}-norm and the initial density are suitably small. Note that the first result is obtained under the condition of ρ0∈Lγ∩W2,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho _{0}\in L^{\gamma }\cap W^{2, q}$\end{document} with q∈(3,6)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q\in (3, 6)$\end{document} and γ∈(1,6)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma \in (1, 6)$\end{document}. It should be noted that the initial total energy can be arbitrarily large, the initial density allowed to vanish, and the system does not satisfy the conservation law of mass (i.e., ρ0∉L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho _{0} \notin L^{1}$\end{document}). Thus, the results obtained particularly extend the one due to Li–Xu–Zhang (Li et al. in SIAM J. Math. Anal. 45:1356–1387, 2013), where the global well-posedness of classical solutions with small energy was proved.


Introduction
One of the important problems in the theory of magnetohydrodynamics (MHD) is that of existence of global solutions to the equations of motion for a viscous compressible fluid. In this paper, we consider the MHD system of equations for a compressible isentropic MHD flows which in the case of 3D motion has the form (cf. [1,7]): (ρu) t + div(ρu ⊗ u) + ∇P(ρ) = μ u + (μ + λ)∇ div u + (∇ × B) × B, B t + u · ∇B -B · ∇u + B div u = ν B, div B = 0, (1.1) where t ≥ 0 is the time, x ∈ R 3 is the spatial coordinate, and ρ ≥ 0, u = (u 1 , u 2 , u 3 ), B = (B 1 , B 2 , B 3 ) are the fluid density, velocity, and magnetic field, respectively. The pressure P = P(ρ) satisfies the condition P(ρ) = Aρ γ with A > 0, γ > 1, (1.2) where γ > 1 is the adiabatic exponent, and A > 0 is a physical constant. The viscosity coefficients μ and λ satisfy Positive constant ν is the resistivity coefficient acting as a magnetic diffusivity of magnetic field. In the second equation in (1.1), the circled times ⊗ means matrix multiplication, namely if a = (a 1 , a 2 , a 3 ), b = (b 1 , b 2 , b 3 ), then Now, we consider the Cauchy problems of (1.1)-(1.3) with (ρ, u, B)(x, t) vanishing at infinity: (ρ, u, B)(x, t) → 0 as |x| → ∞, (1.4) and the initial conditions: (ρ, u, B)(x, 0) = (ρ 0 , u 0 , B 0 )(x) with x ∈ R 3 . (1.5) A great number of works have been devoted to the well-posedness theory of the multidimensional compressible MHD equations. The system of equations (1.1) describes the interaction between fluid flow and magnetic field. If we ignore the magnetic effects in (1.1) (i.e., B = 0), then the MHD system reduces to the Navier-Stokes system, which has been discussed by many mathematicians (see, for example, [11,17,18]). In [5], Huang et al. (2012) established the global existence and uniqueness of classical solutions to the Cauchy problem for the compressible Navier-Stokes equations in 3D with smooth initial data that are of small energy. Then, in [6], Huang et al. (2014) considered the two-dimensional density-dependent Navier-Stokes equations over bounded domains, and they derived a new blow-up criterion for strong solutions with vacuum. In [10], Li et al. (2019) were concerned with the global well-posedness and large time asymptotic behavior of strong solutions to the Cauchy problems of the Navier-Stokes equations for viscous compressible barotropic flows in 2D and 3D. However, if we consider the influence of magnetic field, the physical phenomena and mathematical structure of these equations make it more complex than the Navier-Stokes system, which makes more and more researchers begin to study the equations with magnetic field (see [2,14,16] and the references therein). Moreover, in [3], Fan and Li obtained the global strong solutions to the 3D compressible nonisentropic MHD equations with zero resistivity, and the results do not need the positivity of initial density, thus, it may vanish in an open subset of the domain. Hu and Wang in [4] considered the equations of the three-dimensional viscous, compressible, and heat-conducting magnetohydrodynamic flows in a bounded domain, and they obtained a solution to the initial-boundary value problem through an approximation scheme and a weak convergence method, and then, the existence of a global variational weak solution to the three-dimensional full magnetohydrodynamic equations with large data was established. Later, in [4], they got the global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. In [15], Zhang et al. (2009) studied the initial boundary value problems of MHD equations in plasma physics, and obtained the global existence of weak solutions with cylindrical symmetry. Recently, for the Cauchy problem, Li et al. (2013) in [9] considered the threedimensional isentropic compressible magnetohydrodynamic equations, and they proved the global well-posedness of a classical solution with small energy but possibly large oscillations, where the flow density was allowed to contain vacuum states. Later, Si et al. (2018) in [13] improved the result of [17], and excluded the unsatisfactory restriction on the adiabatic exponent (i.e., γ ∈ (1, 3/2)), and they obtained the global classical solutions of compressible isentropic Navier-Stokes equations with small density and the adiabatic exponent γ ∈ (1, 6) and γ ∈ (1, ∞), respectively.
The main purpose of this paper is to obtain the global existence and uniqueness of classical solution of the problem (1.1)-(1.5). Before stating the main results, we explain the notation and conventions used throughout this paper. We denote For 1 < r < ∞ and k ∈ Z, we denote the standard homogeneous and inhomogeneous Sobolev spaces: The total energy is defined as follows: 6) and the initial energy is denoted by E 0 , i.e., The first result of this paper is formulated in the following theorem.
Theorem 1.1 For any given numbers M 0 , M 1 > 0 and q ∈ (3, 6), suppose that 8) and the compatibility condition holds then for any 0 < T < ∞, there exists a unique global classical solution (ρ, u, B) of the problem for any 0 < τ < T < ∞.
Remark 1.1 In Theorem 1.1, the classical solution of (1.1)-(1.5) is justified under the condition that the initial density and the L 3 -norm of the initial magnetic field are sufficiently small, and this solution is far away from the initial time. It is worth noting that the total initial energy E 0 can be arbitrarily large and the vacuum states are allowed.
Remark 1.2 The proof of Theorem 1.1 is based on a new t-weighted estimate of (∇u, ∇B t ) L 2 (see (3.56)), and the L 1 (0, T; L ∞ )-estimate of the effective viscous flux F will be achieved by making full use of (3.56). It is worth pointing out that the effective viscous flux F plays an important role in applying the Zlotnik's inequality (see Lemma 2.3) to finish the proof of the (a priori) upper bound of the density. Remark 1.3 Indeed, if, in addition, the conservation law of the total mass holds (i.e., ρ(t) L 1 = ρ 0 L 1 for all t > 0), then Theorem 1.1 is similar to the results of [12] for all γ > 1.
The rest of the paper is organized as follows. In Sect. 2, we recall some known facts and elementary inequalities which will be frequently used later. Section 3 is devoted to the global a priori estimates, which are necessary for the proof of Theorem 1.1.

Preliminaries
In this section, we will recall some known facts and elementary inequalities which will be used frequently later. We start with the well-known Gagliardo-Nirenberg inequality [8].
Lemma 2.1 For p ∈ [2, 6], q ∈ (1, ∞), and r ∈ (3, ∞), assume that f ∈ H 1 (R 3 ) and g ∈ L q (R 3 ) ∩ D 1,r (R 3 ). Then there exists a generic constant C > 0, depending only on q and r, such that (2.2) As in [9], we introduce the effective viscous flux F, the vorticity ω, and the material derivative"·", which are defined as follows: Then it is easily derived from (1.1) that Thus, it follows from Lemma 2.1 and the standard L p -estimates of elliptic equations that we have the following lemma.
The proof of Lemma 2.2 can be found in [9,13], hence, we skip it for simplicity.
Finally, we state the local existence result of classical solutions to the problem (1.1)-(1.5) with large initial data which may contain vacuum states (see [9]). (2.12)

Proof of Theorem 1.1
In the section, we will establish the uniform a priori bounds of local solutions (ρ, u, B) to the Cauchy problems (1.1)-(1.5) whose existence is guaranteed by Lemma 2.4. Thus, let T > 0 be a fixed time and (ρ, u, B) be the smooth solution of (1.1)-(1.5) on R 3 × [0, T] with smooth initial data (ρ 0 , u 0 , B 0 ) satisfying (1.8). To estimate this solution, we define Here, (f , g) L p f L p + g L p . The proof of Theorem 1.1 is based on the following key a priori estimates of (ρ, u, B).
The proof of Proposition 3.1 will be presented by a series of lemmas below. For simplicity, we will use the conventions that C and C i (i = 1, 2, . . . ) denote various positive constants, which may depend on μ, λ, ν, γ , A, E 0 , M 1 , and M 2 , but are independent of T and M 0 . Sometimes we also write C(α) to emphasize the dependence on α.
We first begin with the following standard energy estimates, which can be easily deduced from (1.1)-(1.5).
By virtue of (3.1) and (3.4), we infer from Lemma 2.1 (p = 6 in (2.1)) that Here the constant C > 0 comes from the Gagliardo-Nirenberg-Sobolev inequality in (2.1), and we use the constant C(E 0 ) > 0 to emphasize the dependence on E 0 . Thus, combining (3.4) and (3.5) yields the following lemma: where the constant C > 0 depends on ν, E 0 , and the coefficients of the Gagliardo-Nirenberg-Sobolev inequality in Lemma 2.1, but is independent of M 0 .
Proof Multiplying the third equation of (1.1) by 3|B|B and integrating by parts over R 3 , we have where the last term on the right-hand in (3.7) comes from the following inequality: To deal with the right-hand side of (3.7), we notice that which, together with (3.7), yields As a result, we deduce from (3.5) and the Gronwall's inequality that (3.6) holds. Now, to estimate A 1 (T) and A 2 (T), we first prove the following lemma.
We are now in a position of providing the concluding estimates of A 1 (T) and A 2 (T).
In order to derive a uniform upper bound of the density, we still need the following t-weighted estimate.
Proof Indeed, it follows from (3.6), (3.38), and (3.40) that (3.58) Multiplying (3.36) by (tt 1 ), integrating it over (t 1 , t) with t 1 ≤ t ≤ t 2 , using (3.6) and (3.40), we get where the second term on the right-hand side can be estimated as follows based on (3.58): By virtue of (3.57) and Cauchy-Schwarz inequality, we also have Similarly, we infer from (3.40) that In addition, it holds by (1.1) 3 and (3.40) that (3.60) Substituting J 1 , J 2 , and J 3 into (3.59) and using (3.60), we obtain We are now in a position of estimating an upper bound of the density.
Proof In view of (2.3) 1 , we can rewrite (1.1) 1 in the form: Obviously, it holds that g(∞) = -∞. So, to apply Lemma 2.3, we still need to deal with b(t). To do this, we first utilize (2.2), (2.5), (2.6), and the fact that 0 ≤ ρ ≤ 2M 0 to deduce that for any 0 ≤ t 1 The right-hand side of (3.65) can be estimated as follows. We deduce from (3.4), (3.56), and Hölder inequality that Similarly, using (3.56) and Hölder inequality, we have By virtue of (3.6) and (3.40), we infer from Cauchy-Schwarz inequality that With the help of Proposition 3.1, the higher-order estimates of the solution (ρ, u, B) can be shown in a manner similar to that in [9] (see . For completeness and for convenience, we collect these estimates in the following lemma without proofs. With Propositions 3.1-3.2 at hand, we can extend the local solutions obtained in Lemma 2.4 to be a global one in a similar manner as that in [9]. Proof of Theorem 1.1 By Lemma 2.4, there exists a T * > 0 such that the problem (1.1)-(1.5) has a classical solution (ρ, u, B) on (0, T * ]. Noting that A 1 (0) + A 2 (0) = M 1 + M 2 ≤ K, 0≤ ρ 0 ≤ M 0 , and using the continuity arguments, one infers that there exists a T 1 ∈ (0, T * ] such that (3.1) holds for T = T 1 . Next, let T * sup{T| (3.1) holds}. (3.67) Then T * ≥ T 1 > 0. We claim that T * = ∞. where we have also used the following embedding: L ∞ τ , T; H 1 ∩ H 1 τ , T; H -1 → C τ , T; L α , ∀α ∈ [2, 6).
This particularly implies that (ρ, u, B)(x, T * ) satisfies the compatibility condition (1.9) with g(x) u(x, T * ) at t = T * . Thus, using Lemma 2.4, (3.2), and the continuity arguments, we know that there exists a T * * > T * such that (3.1) holds for T = T * * , which contradicts (3.67). Hence, (3.68) holds. This, together with Proposition 3.2 again, shows that the solution (ρ, u, B) is in fact the unique classical solution on R 3 × (0, T] for any 0 < T < T * = ∞. The proof of Theorem 1.1 is therefore complete.