Regularity of large solutions for the compressible magnetohydrodynamic equations

In this paper, we consider the initial-boundary value problem of one-dimensional compressible magnetohydrodynamics flows. The existence and continuous dependence of global solutions in H¹ have been established in Chen and Wang (Z Angew Math Phys 54, 608-632, 2003). We will obtain the regularity of global solutions under certain assumptions on the initial data by deriving some new a priori estimates.

Magnetohydrodynamics (MHD) is concerned with the flow of electrically conducting fluids in the presence of magnetic fields, either externally applied or generated within the fluid by inductive action. The application of magnetohydrodynamics covers a very wide range of physical areas from liquid metals to cosmic plasmas, for example, the intensely heated and ionized fluids in an electromagnetic field in astrophysics, geophysics, high-speed aerodynamics, and plasma physics. There is a complex interaction between the magnetic and fluid dynamic phenomena, and both hydrodynamic and electrodynamic effects have to be considered. For convenience, we consider the following plane magnetohydrodynamic equations in the Lagrangian coordinate system:

Here, v, u, w, b, θ, and p are the specific volume, the longitudinal velocity, the transverse velocity, the transverse magnetic filed, the absolute temperature, and the pressure, respectively; λ, μ, ν, and κ are the bulk viscosity coefficient, the shear viscosity coefficient, the magnetic diffusivity, and the heat conductivity, respectively.

We consider problem (1.1)-(1.5) in the region {y ∈ Ω: = (0, 1), t ≥ 0} under the initial-boundary conditions

In this paper, we focus on an initial-boundary problem for the magnetohydrodynamic flows of a perfect gas with following equations of state:

where R is the gas constant and c_v is the heat capacity of the gas at constant volume. For concreteness, we assume that λ, μ, and ν are constants, and κ depends on the temperature θ with C₁ ≤ κ(θ)/(1 + θ^r) ≤ C₂ for some positive constants C₁, C₂ and, r ≥ 2. The growth condition assumed on κ is motivated by the physical fact: κ ∝ θ ^5/2 for important physical regimes (see [1,2]). The total energy of the magnetohydrodynamics flows is

Before showing our main results, let us first recall the related results in the literature. For the one-dimensional ideal gas, i.e.,

with suitable positive constants c_v, R. Kazhikhov and Shelukhin [3-5], Kawashima and Nishida [6] established the existence of global smooth solutions. Zheng and Qin [7] proved the existence of maximal attractors in Hⁱ(i = 1, 2). However, under very high temperatures and densities, constitutive relations (1.8) become inadequate. Thus, a more realistic model would be a linearly viscous gas (or Newtonian fluid)

satisfying Fourier's law of heat flux

whose internal energy e and pressure p are coupled by the standard thermodynamical relation (1.8). In this case, Kawohl [8] obtained the existence of global solutions with the exponents r ∈ [0, 1], q ≥ 2r + 2. Jiang [9] also established the global existence with basically same constitutive relations as those in [8] but with the exponents r ∈ [0, 1], q ≥ r + 1. When the exponents q, r satisfy the more general constitutive relations than those in [8,9], Qin [10] established the regularity and asymptotic behavior of global solutions with arbitrary initial data for a one-dimensional viscous heat-conductive real gas.

For the radiative and reactive gas, Ducomet [11] established the global existence and exponential decay in H¹ of smooth solutions, and Umehara and Tani [12] proved the global existence of smooth solutions for a self-gravitating radiative and reactive gas.

For the radiative magnetohydrodynamic equations with self-gravitation, Ducomet and Feireisl [13] proved the existence of global-in-time solutions of this problem with arbitrarily large initial data and conservative boundary conditions on a bounded spatial domain in ℝ³. Recently, under the technical condition that κ(ρ, θ) satisfies

for some , Zhang and Xie [14] investigated the existence of global smooth solutions.

For the non-radiative and non self-gravitation magnetohydrodynamic flows, there have been a number of studies under various conditions by several authors (see, e.g., [2,15-22]). The existence and uniqueness of local smooth solutions were first obtained in [21]; moreover, the existence of global smooth solutions with small smooth initial data was shown in [20]. Chen and Wang [15] investigated a free boundary problem with general large initial data with exponents r ∈ [0, 1], q ≥ 2r + 2. Under the technical condition that κ(ρ, θ) satisfies

for q ≥ 2, Chen and Wang [16] also proved the existence and continuous dependence of global strong solutions with large initial data. Wang [22] established large solutions to the initial-boundary value problem for planar magnetohydrodynamics. Under the technical condition upon

Fan et al. [18] investigated the uniqueness of the weak solutions of MHD with Lebesgue initial data. Fan et al. [19] also considered a one-dimensional plane compressible MHD flows and proved that as the shear viscosity goes to zero, global weak solutions converge to a solution of the original equations with zero shear viscosity. The uniqueness and continuous dependence of weak solutions for the Cauchy problem have been proved by Hoff and Tsyganov [17].

As mentioned above, the global existence in of global solutions has never been studied for Equations (1.1)-(1.5) of the nonlinear one-dimensional compressible magnetohydrodynamics flows with initial-boundary conditions (1.6)-(1.7). The main aim of this paper is to prove the regularity of solutions in the subspace of (Hⁱ[0, 1])⁷(i = 2, 4) for systems (1.1)-(1.7). In order to obtain higher regularity of global solutions, there are many complicated estimates on higher derivations of solutions to be involved, this is our main difficulty. To overcome this difficulty, we should use some proper embedding theorems, the interpolation techniques as well as many delicate estimates. This is the novelty of the paper.

We define three spaces as follows:

The notation in this paper will be stated as follows:

L^p, 1 ≤ p ≤ +∞, W^{m, p}, m ∈ N, H¹ = W^1,2, denote the usual (Sobolev) spaces on Ω. In addition, ||·||_B denotes the norm in the space B, we also put . Constants C_i(i = 1, 2, 3, 4) depend on the norm of the initial data (v₀, u₀, w₀, b₀, θ₀) and T > 0.

Now we are in a position to state our main results.

Theorem 1.1 Assume that the initial data and e, p, and κ are C³ functions. Then, the problem (1.1) -(1.7) admits a unique global solution such that for any T > 0,

where , constant is determined by

Theorem 1.2 Assume that the initial data and e, p, and κ are C⁵ functions on 0 < v < +∞ and 0 ≤ θ < +∞. Then, the problem (1.1)-(1.7) admits a unique global solution such that for any T > 0,

In this section, we study the global existence of problem (1.1)-(1.7) in by establishing a series of priori estimates. Without loss of generality, we take c_v = R = 1. We begin with the following lemma.

Lemma 2.1 Assume that the initial data and e, p, and κ are C² functions on 0 < v < +∞ and 0 ≤ θ < +∞ and there exists a positive constant C₀ such that

Then, for the initial data , the problem (1.1) -(1.7) admits a unique global solution such that for any T > 0

and for any t ∈ [0, T],

Proof. See, e.g., [16].

Lemma 2.2 Under the assumptions in Theorem 1.1, the following estimate holds:

Proof. Differentiating (1.2) with respect to t, multiplying the resultant by u_t, and then integrating the resulting equation over Q_t : = Ω × [0, t], we infer

which implies

Analogously, we have

Thus, (2.3) follows from (2.4)-(2.5).

Lemma 2.3 Under the assumptions in Theorem 1.1, the following estimate holds:

Proof. Equation (1.2) can be rewritten as

Using equation (2.7), Lemmas 2.1-2.2, Sobolev's embedding theorem and Young's inequality, we have

which leads to

Similarly, we derive

Thus, (2.6) follows from (2.8)-(2.10).

Lemma 2.4 Under the assumptions in Theorem 1.1, the following estimate holds:

Proof. Differentiating (1.2) with respect to y, we obtain

where

Multiplying (2.12) by , integrating the resulting equation over Q_t, and then using the Young inequality and interpolation theorem, we can conclude

which, together with Lemmas 2.1-2.3, yields (2.11).

Proof of Theorem 1.1. By Lemmas 2.1-2.4, we complete the proof of Theorem 1.1.

In this section, we study the global existence of problem (1.1)-(1.7) in by establishing a series of priori estimates. We begin with the following lemmas.

Lemma 3.1 Under the assumptions in Theorem 1.2, the following estimates hold:

Proof. We easily infer from (1.2), Lemma 2.1 and Theorems 1.1 that

Differentiating (1.2) with respect to y, and using Theorem 1.1, we get

or

Differentiating (1.2) with respect to y twice, using the embedding theorem and Theorem 1.1, we conclude

or

Similarly, we have

or

or

or

or

or

or

Differentiating (1.2) with respect to t, and using Theorem 1.1, (3.3), (3.5), (3.11)-(3.12) and (3.16), we derive

Similarly, we can conclude

Thus, (3.1) follows from (3.3), (3.7), (3.11) and (3.16), and (3.2) from (3.5), (3.9), (3.13), (3.18) and (3.20)-(3.23).

Lemma 3.2 Under the assumptions in Theorem 1.2, the following estimates hold, for any t ∈ [0, T],

Proof. Differentiating (1.2) with respect to t twice, multiplying the resulting equation by u_tt, performing an integration by parts, and using Lemma 2.1, we have

Thus, using Theorem 1.1 and Lemma 3.1, we get

Analogously, we obtain

Equation (1.5) can be rewritten as

Differentiating (3.28) with respect to t twice, multiplying the resulting equation by θ_tt in L² 0[1] and integrating by parts, we arrive at

By virtue of Theorem 1.1 and Lemmas 3.1-3.2, using the embedding theorem, we deduce for any ε ∈ (0, 1),

and

which implies

which implies

Thus, for ε ∈ (0, 1) small enough, we derive from above estimates

Thus, taking supremum in t on the left-hand side of (3.29), picking ε ∈ (0, 1) small enough, and using (3.23), we can derive estimate (3.26).

Lemma 3.3 Under the assumptions in Theorem 1.2, the following estimates hold, for any t ∈ [0,T],

Proof. Differentiating (1.2) with respect to y and t, multiplying the resulting equation by u_ty, and integrating by parts, we arrive at

where

We use Theorem 1.1, Lemma 2.1, the interpolation inequality and Poincaré's inequality to obtain

where

Using the Young inequality several times, we derive

and

Thus, we infer from (3.34)-(3.36) that

which, together with Theorem 1.1, Lemma 2.1, and Lemmas 3.1-3.2, yields

Similarly, by Theorem 1.1, Lemma 2.1, and Lemmas 3.1-3.2 and the embedding theorem, we have

which, combined with (3.33), (3.37)-(3.38), Theorem 1.1, Lemma 2.1, and Lemmas 3.1-3.2, gives that for ε ∈ (0, 1) small enough,

On the other hand, differentiating (1.2) with respect to x and t, using Theorem 1.1 and Lemmas 3.1-3.2, we have

Thus, inserting (3.40) into (3.39) implies estimate (3.30).

Analogously, we can obtain estimates (3.31)-(3.32). □

Lemma 3.4 Under the assumptions in Theorem 1.2, the following estimates hold for any t ∈ [0,T],

Proof. Adding up (3.30)-(3.32), picking ε ∈ (0, 1) enough small, by Lemmas 3.1-3.3, and Gronwall's inequality, we get

Now multiplying (3.24)-(3.26) by ε, ε, and , adding the resultant to (3.45), and choosing ε ∈ (0, 1) small enough, we obtain

which, by Gronwall's inequality, gives the estimate (3.41).

Differentiating (2.12) with respect to y, and using v_tyy = u_yyy, we obtain

with

Obviously, we can infer from Lemmas 3.1-3.3 that

leading to

Multiplying (3.46) by , we get

which, combined with (3.48), gives

By (3.4), (3.6), (3.8), (3.10), (3.12), (3.14), (3.17), (3.19), (3.41), (3.50), and Lemmas 3.1-3.3, and using the embedding theorem, we have

Differentiating (1.2)-(1.5) with respect to t, using (3.41) and Lemmas 3.1-3.3, we get

which, combined with (3.6), (3.10), (3.14) and (3.19), yields

Therefore, it follows from (3.51), (3.56), and the embedding theorem, we obtain

Differentiating (3.46) with respect to y, we obtain

where .

Using the embedding theorem and Lemmas 3.1-3.3, we can conclude

We infer from (3.20)-(3.23) that

which, together with Lemma 3.3, gives

Thus, it follows from (3.40), (3.59), (3.61), and Lemmas 3.1-3.3 that

Multiplying (3.58) by , we get

whence by (3.62),

Differentiating (1.2) with respect to y there times, using Lemmas 3.1-3.3 and Poincaré's inequality, we have

Thus, we conclude from (1.2), (3.56), (3.61), (3.64), and (3.65) that

Similarly, we can deduce from (1.3)-(1.5) that

which, along with (3.51) and (3.66), gives

Finally, using (1.1), (3.50)-(3.56), (3.64), (3.66)-(3.68), and Sobolev's interpolation inequality, we can get the desired estimates (3.42)-(3.44).

Proof of Theorem 1.2. By Lemma 2.1, Lemmas 3.1-3.4, and Theorem 1.1, we complete the proof of Theorem 1.2.

The authors declare that they have no competing interests.

All authors contributed to each part of this work equally.

The work in part was supported by the NNSF of China (No. 11031003) and the Doctoral Innovational Fund of Donghua University (No. BC20101220).

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