Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vaccum

This paper considers the initial boundary problem to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. The global existence and uniqueness of large strong solutions are established when the heat conductivity coefficient $\kappa(\theta)$ satisfies \begin{equation*} C_{1}(1+\theta^q)\leq \kappa(\theta)\leq C_2(1+\theta^q) \end{equation*} for some constants $q>0$, and $C_1,C_2>0$.


Introduction
Magnetohydrodynamics (MHD) studies the dynamics of conducting fluids in an magnetic field. The MHD finds its way in a very wide range of physical objects, from liquid metals to cosmic plasmas, for example, see [3,23,30,33,37]. The governing equations of compressible planar magnetohydrodynamic flows, which implies that the flows are uniform in the transverse directions, take the following form: where the unknowns ρ ≥ 0 denotes the density of the flow, u ∈ R the longitudinal velocity, w ∈ R 2 the transverse velocity, b ∈ R 2 the transverse magnetic field, and e the internal energy, respectively. Both the pressure P and the internal energy e are generally related to the density and temperature of the flow according to the equations of state: P = P (ρ, θ) and e = e(ρ, θ). The parameters λ = λ(ρ, θ) and µ = µ(ρ, θ) denote the bulk and the shear viscosity coefficients, respectively; ν = ν(ρ, θ) is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field, and κ = κ(ρ, θ) is the heat conductivity.
The system (1.1)-(1.5) are supplemented with the following initial and boundary conditions: (u, w, b, θ x )| ∂Ω = 0, (1.6) (ρ, u, w, b, θ)| t=0 = ρ 0 (x), u 0 (x), w 0 (x), b 0 (x), θ 0 (x) , (1.7) where ∂Ω = {0, 1} denotes the boundary of the interval Ω := (0, 1). The conditions (1.6) mean that the boundary is non-slip and thermally insulated. There have been a lot of studies on MHD by physicists and mathematicians due to its physical importance, complexity, rich phenomena, and mathematical challenges. Below we mention some mathematical results on the compressible MHD equations, the interested readers can refer [3,23,30,33,37] for complete discussions on physical aspects. We begin with the one-dimensional case. The existence and uniqueness of local smooth solutions were proved firstly in [42], while the existence of global smooth solutions with small smooth initial data was shown in [25]. The exponential stability of small smooth solutions was obtained in [35,38]. In [13,40], Hoff and Tsyganov obtained the global existence and uniqueness of weak solutions with small initial energy. Under the technical condition that κ(ρ, θ), depending the temperature θ only, i.e., κ(ρ, θ) ≡ κ(θ), satisfies for some q ≥ 2, Chen and Wang [4] proved the existence, uniqueness, and Lipschitz continuous dependence of global strong solutions to the system (1.1)-(1.5) with large initial data satisfying The similar results are obtained in [5,43] for real gas cases. Recently, Fan, Jiang and Nakamura [8] obtained the global weak solutions to the problem (1.1)-(1.5) when the initial data satisfying the condition (1.8) with some q ≥ 1 and Later they [9] obtained the existence, the uniqueness and the Lipschitz continuous dependence on the initial data of global weak solutions to the problem (1.1)-(1.7) when the initial data lie in the Lebesgue spaces. For the multi-dimensional compressible MHD equations, there are also many mathematical results. As mentioned before, Vol'pert and Hudjaev [42] first obtained the local smooth solutions to the compressible MHD equations. Li, Su and Wang [32] obtained the existence and uniqueness of local in time strong solution with large initial data when the initial density has an positive lower bound. Fan and Yu [11] obtained the strong solution to the compressible MHD equations with vacuum. Kawashima [24] obtained the smooth solutions for two-dimensional compressible MHD equations when the initial data is a small perturbation of given constant state. Umeda, Kawashima and Shizuta [41] obtained the decay of solutions to the linearized MHD equations. Li and Yu [34] obtained the optimal decay rate of small smooth solutions. In [14,15], Hu and Wang obtained the global existence of weak solutions to the isentropic compressible MHD equations and variational solutions to the full compressible MHD equations, see also [7,10,47] for related results. Suen and Hoff [39] obtained the global low-energy weak solutions of the isentropic compressible MHD equations. Xu and Zhang [46] obtained a blow-up criterion to the isentropic compressible MHD equations. We mention that the low Mach limit to the compressible MHD equations is an very important topic, and the interested reader can refer [16, 19-22, 28, 29, 31, 36] and the references cited therein.
It should be point out that although there are many progress on compressible MHD equations it is still an open question to obtain the global strong or smooth solutions to the full compressible MHD equations with large initial data and possible vaccum even in the one dimensional case, see [15].
In the present paper we study the global existence and uniqueness of large strong solutions to the planner compressible magnetohydrodynamic equations (1.1)-(1.5) with large initial data and vaccum. We focus on the perfect gas case: where R > 0 is the gas constant and C V > 0 is the heat capacity of the gas at constant volume. We will consider the case that the coefficients λ, µ, and ν are positive constants and the heat conductivity coefficient depending the temperature θ only, i.e., κ(ρ, θ) ≡ κ(θ). Since the positive physical constants λ, µ, ν, R, and C V do not create essential mathematical difficulties in our analysis, we normalize them to be one for notational simplicity.
The main result in this paper reads as follows.
for some g 1 , g 2 , g 3 ∈ L 2 (Ω). Then for any T > 0 there exists a unique global solution (ρ, u, w, b, θ) to the problem (1.1)-(1.7) such that There are two ingredients in our result comparing with the previous results on one-dimensional MHD equations mentioned above. First, in our result the initial density may contains the vaccum provided that it satisfies the compatible conditions (1.9). Next, a relaxed condition on the heat conductivity coefficient is permitted. In fact, in (1.8), we only need q > 0 while in [4,5,43] required q ≥ 2 and in [8] with q ≥ 1. Remark 1.1. It is possible to extend our results to the one-dimensional compressible MHD system with more general state of equations: with some additional assumptions.
When there is no vaccum initially, we can improve the results in [8] to the case q > 0 by applying the arguments developed here.
We remark that when taking w = b = 0 the system (1.1)-(1.7) reduces to the well-known one-dimensional full Navier-Stokes equations and there are a lot of studies on this system. In the case of that the initial density is bounded away from zero, Kazhikhov and Shelukhin [27] first obtained the global smooth solutions for large initial data three decades ago,see also [1,26,48,49] for different extensions. Recently, Huang and Li [18] obtain the global smooth solutions to the full Navier-Stokes system with possible vaccum and large oscillations provided that the total initial energy is sufficient small. Wen and Zhu obtained the global smooth solutions to the one-dimensional full Navier-Stokes system [44] and symmetric higher dimensional full Navier-Stokes system [45] with large initial data. For the variational or weak solutions to the full Navier-Stokes system, see [2,12].
We give a few words on the strategy of the proof. Since the initial data may contains the vaccum we first construct the regularized initial density ρ 0δ (x) = ρ 0 +δ for any δ > 0. Next, for each fixed δ, we can obtain the local and uniqueness existence of strong solutions. Third, we establish sufficient a priori estimates uniformly with δ. Combining the local existence result, and the uniformly a priori estimates, we obtain the desired global existence result by taking the limiting as δ → 0 + and applying standard continuity argument. We remark that the key point in the whole proof is to obtain uniformly a priori estimates where some ideas developed in [44,45] are adapted. Comparing with [44], the main additional difficulties are due to the presence of the magnetic field and its interaction with the hydrodynamic motion of the flow of large oscillation. We shall deal with the terms involving the magnetic field very carefully, see especially Lemmas 2.5-2.7 below.
Before leaving this introduction we recall the following auxiliary inequalities.
Then for any r > 0, there exists a positive constant

Proof of Theorem 1.1
In this section we will prove Theorem 1.1 by consider the initial density ρ 0δ = ρ 0 + δ, as mentioned before, to get a sequence of approximate solutions to (1.1)-(1.7), then taking δ → 0 + after making some a priori estimates uniformly for δ. Since the proof of the local existence and uniqueness of strong solutions to the approximate problem is now standard [6,11], thus we only need to establish the uniform estimates.
Below we still use (ρ, u, w, b, θ) to denote the smooth solutions of approximate problem to (1.1)-(1.7). We shall denote Q T := Ω × [0, T ] with T > 0 and omit the spatial domain Ω in integrals for convenience. We use C to denote the constants which are independent of δ and may change from line to line.
To begin with the proof, we notice that the total mass and energy in the system (1.1)-(1.5) are conserved. In fact, by rewriting (1.1)-(1.5) one has where E and S are the total energy and the entropy, respectively, Integrating (1.1), (2.1) and (2.2) over Q T , we have Proof. We need only to estimate the upper bound. Here we borrow the proof from [8] for reader's convenience. From (1.1) and (1.2), we have By virtue of Lemma 2.1 and Cauchy inequality, it holds Now, denoting F := e φ and using (2.4), we have after a straightforward calculation that D t (ρF ) := ∂ t (ρF ) + u∂ x (ρF ) = − p + 1 2 |b| 2 ρF ≤ 0, which together with (2.5) implies (2.3) immediately.
Lemma 2.3. Let 0 < α < min{1, q} be any given constant, then it holds Proof. The proof is similar to that given in [45] for symmetric Navier-Stokes equations, we present it here for completeness. Multiplying (1.5) by θ −α and integrating the result over Q T , we have Using (1.1) and Lemma 2.1, the first two terms on the right-hand side of (2.8) can be bounded as By Cauchy inequality, Lemmas 2.1 and 2.2, we have (2.10) Noticing 0 < α < min{1, q}, the Hölder inequality and Lemma 1.2 imply that Putting ( Proof. Multiplying (1.2) by u, using (1.1), and integrating the result over Ω, we see that 1 2 (2.14) Similar, multiplying (1.3) by w, using (1.1), and integrating the result over Ω, we obtain 1 2 Multiplying (1.4) by b and then integrating them over Ω, we infer that , and the condition that 0 < α < min{1, q}, we have Integrating (2.17) over [0, T ] and applying the above inequality give (2.13).
Proof. Multiplying (1.2) by u t , and then integrating them over Ω, we infer that First, by Cauchy inequality and Poincaré inequality, it is easy to find that (2.20) From (1.4) and (1.5), we have Multiplying (1.3) by w t , using (2.3), and integrating the result over Ω, we derive Multiplying (1.4) by b t − b xx , integrating the result over Ω, and using Cauchy inequality, we have Multiplying (1.5) by θ q+1 , using (1.8), and integrating the result over Ω, we find that where we used the estimate: Proof. Applying the operator ∂ x to (1.1) gives Multiplying the above equation by 2ρ x , integrating the result over Ω, and using (2.3) and (2.18), we find that On the other hand, it follows from (1.2) that Proof. Applying ∂ t to (1.2), we see that Multiplying the above equation by u t , integrating them over Ω, and using (1.1), Lemmas 2.5 and 2.6, and Cauchy inequality, we obtain 1 2 for any 0 < ǫ 1 < 1.
Applying the operator ∂ t to (1.4) gives Multiplying the above equation by b t , integrating the result over Ω, and using Lemma 2.5, we find that where we have used the following estimate: Noting the above estimate, (1.5), and (2.38), it follows that (2.39) Lemma 2.8.
Proof. Applying the operator ∂ 2 x to (1.1) gives Multiplying the above equation by 2ρ xx , integrating them over Ω, and using Lemmas 2.7 and 2.6, we find that Applying ∂ x to (1.2), integrating them over Ω, and using Lemmas 2.6 and 2.7, we infer that Inserting the above estimates into (2.41), and using Lemma 2.7 and the Gronwall inequality, we get Since Finally, noting Lemma 2.9.