Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data

The free boundary problem of planar full compressible magnetohydrodynamic equations with large initial data is studied in this paper, when the initial density connects to vacuum smoothly. The global existence and uniqueness of classical solutions are established, and the expanding rate of the free interface is shown. Using the method of Lagrangian particle path, we derive some L ∞ estimates and weighted energy estimates, which lead to the global existence of classical solutions. The main difficulty of this problem is the degeneracy of the system near the free boundary, while previous results (cf. [ 4 , 30 ]) require that the density is bounded from below by a positive constant.


1.
Introduction. Magnetohydrodynamics (MHD) concerns the motion of conducting fluids in an electromagnetic field. The applications of MHD cover a broad range of physical areas, for example, liquid metals and cosmic plasmas. The dynamics of MHD are affected obviously by the magnetic field, thus the hydrodynamic and electrodynamic effects are strongly coupled. The planar magnetohydrodynamic flows with magnetic diffusion are governed by the following MHD equations (cf. [4]) ρ t + (ρu) η = 0, η ∈ R, t > 0, (ρu) t + (ρu 2 + p + 1 2 |b| 2 ) η = (λu η ) η , where η, t, ρ, u ∈ R, w ∈ R 2 , b ∈ R 2 , T denote, respectively, the spatial variable, time variable, density of the flow, longitudinal velocity, transverse velocity, the transverse magnetic field, and the temperature. The longitudinal magnetic field is a constant which is taken to be unity in (1.1). The total energy of the planar 538 YAOBIN OU AND PAN SHI magnetohydrodynamic flows is where both the pressure p and the internal energy e are related to the density and temperature of the flow via the equations of states for perfect gases: Here R is the generaic gas constant and c v is the specific heat per volume. In what follows, c v and R are set to be unity for convenience. The constants λ and µ are the viscosity coefficients of the flow, ν is the magnetic diffusivity constant, κ = κ(T ) is the heat conductivity coefficient, and all these coefficients are independent of the magnitude and direction of the magnetic field. We are interested in a free-boundary problem of (1.1) for η ∈ I(t), t > 0 with the following initial-boundary conditions: (ρ, u, w, b, T )| t=0 = (ρ 0 , u 0 , w 0 , b 0 , θ 0 )(η), η ∈ I(0) = (0, 1), (w, b, T η )| ∂I(t) = 0, u| η=0 = 0, (p − λu η )| η=Γ(t) = 0, (1.3) where I(t) = (0, Γ(t)) is the free interval occupied by the fluids with the boundary ∂I(t) = {0, Γ(t)}, and Γ(t) represents the moving interface between the fluid and vacuum with speedΓ(t).
The free boundary problems of compressible fluids, which arise in several important physical situations such as astrophysics, shallow water waves, magnetohydrodynamics and so on, have been intensively studied in the literature.
When the density of fluid is always bounded from below by a positive constant, the problem is extensively studied and many important progresses are achieved during the past years. The readers may refer to [2,11,12,19,27,28,29,33,35,38,43] for the results of Navier-Stokes equations, for instance. For planar MHD equations, the global solution was established in [4] when the initial datum was in H 1 , and the global well-posedness of classical solutions was studied in [30]. Related initial-boundary value problems for planar MHD equations in Lagrangian coordinates are studied in [15,31].
When the fluid connects to vacuum states continuously at the free interface, strong singularity occurs near the free boundary. This case is the so-called vacuum free boundary problem. There have been some local-in-time results for weak solutions of compressible Navier-Stokes equations (cf. [21,39]) and for smooth solutions of various compressible hydrodynamic equations (cf. [5,6,16,17,18,14,22]). Concerning the global-in-time results, most of the known results are for weak solutions or strong solutions, in particular for Navier-Stokes equations. Please refer to [3,7,8,9,13,20,23,26,34,40,44] for instance. However, to have a better understanding on the behaviors of solutions to free boundary problems, it is very important to achieve the global-in-time regularity. For example, the velocity is required to be Lipschitz continuous so that the interface separating fluids and vacuum states is well-defined in the classical sense. The well-posedness of global smooth solutions to vacuum free boundary problems has been established very recently in [24,42] for isentropic flows. However, to our best knowledge, the global existence of classical solution for vacuum free boundary problems of MHD equations, especially in the non-isentropic regime, is still open, due to the difficulties from the combined effects of the temperature, magnetic field and the degeneracy of the system near the free boundary.
In this paper, we show the global existence and uniqueness of classical solutions to the vacuum free boundary problems of full compressible planar MHD equations with large initial data. Since the density always connects to vacuum smoothly, the system is degenerate near the free boundary, while previous results [4,30] require that the density of fluid is strictly positive at the free surface. We remark that the method in this paper also applied to the situations in [4,30].
In the aforementioned studies of vacuum free boundary problems for viscous flows, the method of Lagrangian mass coordinates was often used, by which the global existence of weak solutions was given. However, in order to trace the free boundary, the regularity of velocity field is usually required to be Lipschitz with respect to spatial variables, for instance. But it is hard to derive this estimate in the case that the system is degenerate near the free boundary. As in [5,6], we use a method of Lagrangian trajectory, so that the problem can be transform into an equivalent initial-boundary problem in the usual sense. Based on the conservation of total energy, we first show the positive upper and lower bounds of η x (x, t), where the Lagrangian variable η(x, t) is defined in (2.1). These crucial estimates ensure the equivalency of the original free boundary problem and resulting initial-boundary value problem. For the isentropic Navier-Stokes equations, η x (x, t) can be estimated straightforwardly in terms of initial energy (cf. [42]), however, for the non-isentropic magnetohydrodynamics, there are some essential difficulty in the crucial estimates for η x and η −1 x , due to the combined effects of the temperature and magnetic field. We overcome the trouble by showing first the estimates for I η x (x, t)dx (indeed, it give the expanding rate for the free interval), and then some new estimates for the temperature and magnetic field, that is, the bounds for t 0 θ(·, s) L ∞ ds and t 0 |B| 2 (x, s)ds. Next, we show the weighted energy estimates for lower order derivatives of the solutions, which can give the global existence of strong solutions to (1.1)-(1.3). In particular, we obtain the estimate for v x (·, t) L ∞ , which guarantees the regularity of particle path η(x, t) up to the boundary. Finally, we estimate the higher order derivatives of solutions by delicate analysis, so that the estimates for each term in (1.1) can be improved to be non-weighted and the global existence of smooth solutions for (1.1)-(1.3) is shown. The non-weighted estimates are based on an inequality obtained in [36].
The rest of this paper is organized as follows. In Section 2, we will reformulate the free boundary problem (1.1)-(1.3) by the method of Lagrangian particle path and state the main results of this article. Section 3 will be devoted to some preliminaries. In Section 4, we derive the uniform estimates for the solutions, which gives the global existence of the classical solution.
2. Lagrangian reformulation and main results. The main idea to solve the problem (1.1)-(1.3) is to transform it into an equivalent initial-boundary value problem with fixed boundary. To this end, we use x as the reference variable and define the Lagrangian variable η(x, t) by η t (x, t) = u(η(x, t), t) for t > 0 and η(x, 0) = x, x ∈ I. (2.1) For simplicity of presentation, the initial domain is denoted as I := (0, 1). We set the Lagrangian density, longitudinal velocity, transverse velocity, temperature, and magnetic field, respectively, by for any (x, t) ∈ I × [0, T ]. We consider the situation that the fluid always occupy the region I(t), and the density vanishes at the free boundary Γ(t). Then the Lagrangian version of (1.1) can be written equivalently as and θ 0 ≥ θ 0 > 0 in I ∪ ∂I, whereρ 0 and θ 0 are given positive constants. We should note that η x > 0 for all (x, t) ∈ I × [0, T ], which can be verified in the later estimate. (2.4) Thus the system (2.3) can be rewritten as It is well known that, for the heat-conductive fluids, κ is a non-decreasing function of the temperature θ. In particular, κ(θ) ≈ 1+θ q , q > 0 for some important physical regimes (see [1,41]), for instance, q ∈ (4.5, 5.5) for the fluids with high temperature (cf. [10]). In this paper, we assume that the heat conductivity coefficient κ(θ) satisfies κ(θ) ∈ C 3 for any θ > 0, where q, c i (i = 1, 2) are positive constants. Now, we are ready to state the main result of this paper.
The proof of this theorem is given in Section 4.
Remark 3.2. The multi-dimensional version of Lemma 3.1 can be found in [10]. Moreover, when q = 1 and ρ 0 (x) = constant, this lemma reduces to the Poincaré inequality. [32,37]) Suppose that X ⊂ Z ⊂ Y are Banach spaces and the embedding X ⊂ Z is compact.

Notation.
1) Throughout the rest of paper, C will denote a positive constant which does not depend on the data, but possibly on the given constant T . They are referred as universal and can change from one inequality to another one. Also we use C(δ) to denote a certain positive constant depending on the quantity δ.
2) We will employ the notation a b to denote a ≤ Cb, where C is the universal constant as defined above.
3) In the rest of the paper, we will use the notations for any 1 ≤ p ≤ +∞ and 1 ≤ k < +∞.
4. Proof of Theorem 2.1. The local existence and uniqueness to (2.5) can be established similarly as in [5] by the method of finite difference scheme, thus we omit the details here. To show the global existence of strong or classical solutions, we only need to obtain uniform-in-time estimates with regularity, which is the main task of this section. Suppose that (v, W, B, θ) is the solution to (2.5) in I × [0, T ], which belong to the classes of functions in (2.11). Moreover, we assume which will be recovered later in Lemma 4.3.

4.1.
Lower order estimates. In this subsection, we establish the uniform estimates in the part (i) of Theorem 2.1.
where C is a positive constant depending on η x and v x . Thus one gets using the non-positiveness of H(θ) and the assumption θ 0 ≥ 0. It follows that where C,C,C are positive constants independent of T . Proof.
Step 1. Lower bounds of η x (x, t) and η(x, t). By (4.1) and the boundary conditon (2.5) 6 , we find Integrating (2.5) 1 from x to 1 yields Next, we also integrate the above equation in t and use the initial condition that (4.7) In virtue of the fact that p = ρ 0 θη −1 x ≥ 0, |B| 2 ≥ 0, and Lemma 4.1, one gets (4.8) Step 2. Upper bounds of η x (x, t) and η(x, t). Let Then with the help of (4.7), we have It follows from (4.2), (4.7) and (4.9) that (4.10) Thus by Gronwall's inequality Moreover, (4.11) It follows that for any ( Now it suffices to evaluate t 0 θ(·, s) L ∞ ds and t 0 |B| 2 ds, which is discussed in two cases as follows.

YAOBIN OU AND PAN SHI
Then we integrate this equation over I × (0, t), using (4.2), (4.8) and the fact δ ∈ (0, 1), to get (4.17) Here the constant C(α) is independent of δ. Thus by Lemmas 3.1 and 4.1, and (4.17) , we have Note that 0 < 1 − α < 1 and 0 < 1 + α − q < 1, and take α to be q/2. Then we choose = (t) to be small enough and use the Young inequality, then let δ → 0 to derive Similarly to (4.16), we derive   In what follows, we always denote by C a generic constant depending possibly on T but not on the unknowns, for the sake of simplicity.

YAOBIN OU AND PAN SHI
Choosing δ < λ/4 and using (4.2), (4.4) and (4.21), we obtain Integrating (2.5) 1 from x to 1, we get Thus we obtain by using (4.2) and (4.21) (4.27) Next, we integrate v t · (2.5) 1 over I to get (4.28) We estimate J 1 and J 2 respectively as follows: and by (2.5) 1 and (2.5) 4 , Moreover, Then it follows from (4.27), (4.28) and the above calculations that On the other hand, we multiply (2.5) 4 by θ 0 κ(y)dy and integrate to get (4.30) From (4.2), (4.29), (4.30), (4.21), (4.25) and the Gronwall inequality, we have (4.31) Similar to the procedure of (4.28), we derive (4.32) We evaluate the two terms on the right hand side as follows: Thus we apply the Gronwall inequality to (4.32) and use (4.2) and (4.4) to get (4.33) To absorb the first term on the right side, we multiply (2.5) 3 by B t and integrate to obtain (4.34) Clearly, Applying the Gronwall inequality to (4.32) and using the above estimates, we obtain Finally, we combine (4.35) and (4.33) and employ the Gronwall inequality of integral type to get (4.39) Proof. Differentiating (2.5) 1 with respect to t, we obtain Then we multiply the above equation by v t and integrate over I to get and Therefore, submitting (4.42) and (4.43) into (4.41), we have by using (4.23), (4.24) and the Gronwall inequality. On the other hand, we integrate κ(θ)θ t · (2.5) 4 over I to get where

YAOBIN OU AND PAN SHI
Then it is easy to show that From Lemmas 4.6, 4.7, 4.8 and 4.9, we obtain the following result.
This proposition gives the energy estimates for the part (i) of Theorem 2.1.

Higher order estimates.
In what follows, we will deal with the higher order estimates for the solutions, so that the system (2.5), is satisfied in the classical sense.
Lemma 4.11. For any t ∈ [0, T ], we have (4.69) Proof. We first multiply (4.50) by W tt and integrate over I × (0, t) to get where and Choosing δ to be small enough, we get Next, we differentiate (2.5) 3 with respect to t and multiply the resulting equation by B tt , then integrate it over I × (0, t) to get Observing that and using Lemmas 4.6 and 4.7, we get where the estimates in Proposition 4.1 are used.

YAOBIN OU AND PAN SHI
Then, we multiply (4.40) by v tt and integrate over I to get We estimate M 1 through M 4 as follows.
Note that by using (2.5) 1 and (2.5) 4 , we have Finally, (4.75) Proof. Applying ∂ t to (4.50), we get (4.76) Multiply the above equation by W tt and integrate over I × [0, t] to get Applying ∂ tt to (2.5) 3 , we get (4.78) Integrate B tt · (4.78) over I, we have So, we get Applying ∂ t to (4.40), we obtain (4.80) We multiply the above equation by v tt and integrate to get which yields Note that Integrating (κθ t ) t · (4.57) over I × [0, T ]. One gets We estimate each term as follows.
Noting that (κθ x ) t = (κθ t ) x , we integrate by parts to get
Utilizing the Gronwall inequality, we have v xxx L 2 (t) ≤ C, t ∈ [0, T ], which gives immediately We differentiate (2.5) 2 twice with respect to x to get (4.86) Then, we have
Differentiating (2.5) 2 twice with respect to x, we have Similarly, we get H 4 estimate of B through