Abstract
In this paper, we consider the initialboundary value problem of onedimensional compressible magnetohydrodynamics flows. The existence and continuous dependence of global solutions in H^{1 }have been established in Chen and Wang (Z Angew Math Phys 54, 608632, 2003). We will obtain the regularity of global solutions under certain assumptions on the initial data by deriving some new a priori estimates.
Keywords:
magnetohydrodynamics (MHD); global solutions; regularity; initialboundary value problem1 Introduction
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, highspeed 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 initialboundary conditions
In this paper, we focus on an initialboundary 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 C_{1 }≤ κ(θ)/(1 + θ^{r}) ≤ C_{2 }for some positive constants C_{1}, C_{2 }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 onedimensional ideal gas, i.e.,
with suitable positive constants c_{v}, R. Kazhikhov and Shelukhin [35], Kawashima and Nishida [6] established the existence of global smooth solutions. Zheng and Qin [7] proved the existence of maximal attractors in H^{i}(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 onedimensional viscous heatconductive real gas.
For the radiative and reactive gas, Ducomet [11] established the global existence and exponential decay in H^{1 }of smooth solutions, and Umehara and Tani [12] proved the global existence of smooth solutions for a selfgravitating radiative and reactive gas.
For the radiative magnetohydrodynamic equations with selfgravitation, Ducomet and Feireisl [13] proved the existence of globalintime solutions of this problem with arbitrarily large initial data and conservative boundary conditions on a bounded spatial domain in ℝ^{3}. 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 nonradiative and non selfgravitation magnetohydrodynamic flows, there have been a number of studies under various conditions by several authors (see, e.g., [2,1522]). 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 initialboundary 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 onedimensional 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 onedimensional compressible magnetohydrodynamics flows with initialboundary conditions (1.6)(1.7). The main aim of this paper is to prove the regularity of solutions in the subspace of (H^{i}[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, H^{1 }= 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 (v_{0}, u_{0}, w_{0}, b_{0}, θ_{0}) 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 C^{3 }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 C^{5 }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,
2 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. 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 C^{2 }functions on 0 < v < +∞ and 0 ≤ θ < +∞ and there exists a positive constant C_{0 }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 2.12.2, 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 2.12.3, yields (2.11).
Proof of Theorem 1.1. By Lemmas 2.12.4, we complete the proof of Theorem 1.1.
3 Proof of Theorem 1.2
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 L^{2 }0[1] and integrating by parts, we arrive at
By virtue of Theorem 1.1 and Lemmas 3.13.2, 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 lefthand 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 3.13.2, yields
Similarly, by Theorem 1.1, Lemma 2.1, and Lemmas 3.13.2 and the embedding theorem, we have
which, combined with (3.33), (3.37)(3.38), Theorem 1.1, Lemma 2.1, and Lemmas 3.13.2, 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 3.13.2, 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 3.13.3, 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 3.13.3 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 3.13.3, and using the embedding theorem, we have
Differentiating (1.2)(1.5) with respect to t, using (3.41) and Lemmas 3.13.3, 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
Using the embedding theorem and Lemmas 3.13.3, 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 3.13.3 that
Multiplying (3.58) by , we get
whence by (3.62),
Differentiating (1.2) with respect to y there times, using Lemmas 3.13.3 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 3.13.4, and Theorem 1.1, we complete the proof of Theorem 1.2.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors contributed to each part of this work equally.
Acknowledgements
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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