Regularity and uniqueness for the 3D compressible magnetohydrodynamic equations

In this paper, some new L p gradient estimates are justiﬁed for the three-dimensional compressible magnetohydrodynamic equations in the whole space R 3 . The key to derive the estimate (cid:2)∇ u (cid:2) L 3 is the “div-curl” decomposition technique. For regular initial data with small energy, we prove the existence of global solutions belonging to a new class of functions in which the uniqueness can be shown to hold. MSC: 35B65; 35Q35; 76D03; 76W05


Introduction
We are concerned with the Cauchy problem of three-dimensional compressible magnetohydrodynamic (MHD) equations which read as (cf. [2,16]) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ρ t + div(ρu) = 0, (ρu) t + div(ρu ⊗ u) + ∇P(ρ) = μ u + (μ + λ)∇ div u + (∇ × B) × B, B t -∇ × (u × B) = -∇ × (ν∇ × B), div B = 0, (1.1) where t > 0, x = (x 1 , x 2 , x 3 ) ∈ R 3 , the unknown functions ρ, 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 is the adiabatic exponent and A is a physical constant. The positive constant ν is the resistivity coefficient, and the viscosity coefficients μ and λ satisfy Let us consider the Cauchy problem of (1.1)-(1.3) with the far-field behavior (ρ, u, B)(x, t) → (1, 0, 0) as |x| → ∞, t > 0, (1.4) and the initial conditions (ρ, u, B)(x, 0) = (ρ 0 , u 0 , B 0 )(x) with x ∈ R 3 . (1.5) Without losing generality, let us assume that the far-field state of density at infinity is equivalent to 1. This set of equations (1.1) describes the interaction between fluid flow and magnetic field, which has been studied by many literature works [3,4,6,7,9,22,23]. For the local strong solutions to the compressible MHD flows, Vol'pert and Khudiaev in [20] got the local strong solutions under the conditions of large initial data and positive initial density. Later, Fan and Yu in [10] extended Vol'pert and Khudiaev's results as the initial density may contain vacuum. In [18], Lu and Huang investigated the 2D full compressible MHD equations with zero heat-conduction and obtained a local strong solution as the initial density and initial magnetic field decay not too slow at infinity. For the global solutions, Fan and Li in [8] investigated the 3D compressible non-isentropic MHD flows with zero resistivity and got the global strong solutions which do not need the positivity of initial density. In [12], Hu and Wang got the existence of a global variational weak solution to the three-dimensional full magnetohydrodynamic flows. Later, in [13], they also got the global existence and large-time behavior of solutions to the 3D equations of compressible MHD flows. In [22], Zhang, Jiang, and Xie obtained the global existence of weak solutions with cylindrical symmetry to the initial boundary value problems of MHD equations in plasma physics.
The global existence and uniqueness of strong solutions to Cauchy problem (1.1) are much subtle and remain open. Therefore, the main purpose of this paper is to investigate the global existence and uniqueness of solutions to (1.1)-(1.3). Lv, Shi, and Xu in [19] studied the Cauchy problem of 2D or 3D compressible MHD flows with vacuum as far-field density and obtained the global existence and uniqueness of strong solutions under the conditions of small total energy. Li, Xu, and Zhang [17] proved the global well-posedness of classical solution to problem (1.1)-(1.5) provided that the initial energy is suitably small. However, the results obtained in [17,19] also required the following compatibility conditions: Roughly speaking, when the density contains vacuum, condition (1.6) implies Clearly, one can use (1.10) to derive the higher-order estimates of the solutions in the previous articles. Similar to the compressible Navier-Stokes equations (see, for example, [11,15]), we introduce the effective viscous flux F, the vorticity ω: Then it is easily derived from (1.1) that To sum up, can the condition of (1.9) be weakened so far when exploring the global solutions of the MHD equations? When the magnetic field in (1.1) was replaced by the temperature, Xu and Zhang in [21] obtained a global "intermediate weak" solution in the nonvacuum case with lower regularity. Thus, our main aim in this paper is to prove the following theorem of solutions with lower regularity than that in (1.9). Theorem 1.1 For any given number p ∈ [9/2, 6), suppose that the initial data (ρ 0 , u 0 , B 0 ) satisfy Then there exists a positive constant ε > 0, depending on μ, λ, ν, then Cauchy problem (1.1)-(1.5) has a global solution (ρ, u, B) in R 3 × (0, ∞) satisfying, for any 0 < T < ∞, Unfortunately, the first term on the right-hand side cannot be bounded by Lemmas 2.1-2.5. So, another way is to use the "div-curl" decomposition technique, that is, To do this, we first operate "div" and "curl" to both sides of (1.1) 2 , and then multiply | div u| div u and | curl u| curl u, respectively, and integrate them over R 3 , which yields where the symbol R denotes the terms which can be absorbed/bounded by the left-hand side and the estimates obtained in Lemmas 2.1-2.5. Due to the lower regularity of initial velocity, u 0 ∈ H 1 ∩ W 1,3 , but u 0 / ∈ H 2 , thus the estimate u t ∈ L ∞ (0, T; L 2 ) ∩ L 2 (0, T; H 1 ) cannot be obtained. In order to deal with the right two terms of the inequality, we use the following equality to substitute the expression of u t into the integrals: Here, we need inf ρ(x, t) > 0, in other words, we need inf ρ 0 > 0. The consequence of doing so is that some new terms appear, but these terms can be controlled by the estimates obtained in Lemmas 2.2-2.5. To sum up, we need the initial density to exclude the vacuum state, and then the estimate ∇u L 3 can be well controlled, which is essentially and technically needed for the analysis of uniqueness.
The rest of the paper is organized as follows: in the next section, Sect. 2, based on the lower-order estimates achieved in [17], we obtain the necessary a priori estimates on strong solutions. Then finally the main result, Theorem 1.1, is proved in Sect. 3.

Preliminaries
In this section, we establish some necessary a priori bounds for smooth solutions to Cauchy problem (1.1)-(1.5). Before stating our main results, we first let (ρ, u, B) be a smooth solution of (1.1)-(1.5) on R 3 × [0, T] for some 0 < T < ∞. We can check that the equations of (1.1) can be written as follows: We start with the following estimates, which have been achieved in [17, Proposition 3.1], thus we do not explain in detail.

Lemma 2.1 For given constants M
There exist positive constants K and ε, depending on μ, λ, ν, A, γ ,ρ, and M, such that if where G(·) is the potential energy density given by then the following estimates hold: Remark 2.1 Estimates (2.4)-(2.7) are independent of time T and the lower bound of density. Furthermore, if the initial density possesses a positive lower bound, then it follows from the expression of G(·) that In the following, we will use the convention that C or C i (i = 1, 2, . . .) denotes a generic positive constant depending on μ, λ, ν, γ , A,ρ, the initial data, and T.
First, we will prove the following refined t-weighted estimates of the material derivative and the gradient of magnetic.

Lemma 2.2 Under the conditions of (2.2) and (2.3), there exists a positive constant C depending on T such that
Proof In order to prove (2.9), we first need to apply tu j [∂ t + div(u·)] to the both sides of the jth equation of (2.1) 2 , then integrate by parts over R 3 , and add the results together. We get by some calculations that (2.10) Now, we estimate the right-hand side of terms of (2.10). Due to the Cauchy-Schwarz inequality, we get by integration by parts that and In order to estimate I 3 , we notice that which, together with (2.4), yields (2.14) Next, for I 4 , we obtain after using the integration by parts that where we have used the Gagliardo-Nirenberg inequality L 2 , ∀v ∈ H 1 and 2 ≤ p ≤ 6. (2.16) By using (2.16) and integrating by parts, we obtain (2.17) What is left is to estimate the term ∇B t L 2 . To this end, noticing that and using the fact that u t =u -u · ∇u, we obtain after direct computations that Now, we estimate J i as follows. By using (2.16) and the Cauchy-Schwarz inequality, we get and and (2.23) Thus, integrating the resulting equations (2.18) and (2.23) over (0, T) and using (2.3), (2.6), and (2.7), we deduce after adding them together that (2.24) We are now in a position of estimating the last two terms on the right-hand side of (2.24). Indeed, it follows from (1.2), (2.4) and the L p -estimates that which, together with (2.7), gives On the other hand, it follows from (2.1) 3 that It follows from (2.7), (2.24), and (2.27) that (2.28) In view of (2.7), (2.6), and (2.26), we have where δ > 0 is an undetermined number.
Next, we need to estimate ∇u L p with 1 < p < ∞ and p = ∞, respectively. Clearly, since it holds that -u = ∇ div u -∇ × curl u, thus, for 1 < p < ∞, we have However, for p = ∞, the above inequality cannot work. Thus, the following Beale, Kato, and Majda type inequality (cf. [1,14]) will be used later to estimate ∇u L ∞ .

Lemma 2.4
For any k ∈ Z + and p ∈ (1, +∞), let D k,p {v ∈ L 1 loc |∂ k v ∈ L p } and D 1 D 1,2 be the homogeneous Sobolev spaces. Then, for any v ∈ D 1 ∩ D 2,p with p ∈ (3, +∞), there exists a positive constant C(p) > 0 such that, for all ∇v ∈ L 2 ∩ D 1,p , (2.34) With the help of Lemmas 2.3 and 2.4, we can now derive the L 2 -L p -estimates (3 < p < 6) of the gradient of density.
The following technical lemma is concerned with the estimate of ∇u L 3 , which plays an essential role in the entire analysis. (2.49) Proof In terms of the standard L p -estimate, to bound ∇u L p with 1 < p < ∞, it suffices to show that both div u L p and curl u L p are bounded. To do this, we first operate div and curl to both sides of (2.1) 2 to get that (2.51) We shall divide the proofs into three steps.
Step I. Estimation of div u L 3 .