Global Solutions of Nonlinear Magnetohydrodynamics with Large Initial Data

Gui-Qiang Chen Department of Mathematics, Northwestern University Evanston, IL 60208, USA Email: gqchen@math.nwu.edu

and

Dehua Wang Department of Mathematics, University of Pittsburgh Pittsburgh, PA 15260, USA Email: dwang@math.pitt.edu.

Version: October 24, 2001

A free boundary problem for nonlinear magnetohydrodynamics (MHD) with general large initial data is investigated. The existence, uniqueness, and regularity of global solutions are established with large initial data in H^{1}. It is showed that neither shock waves nor vacuum and concentration in the solutions are developed in a finite time, although there is a complex interaction between the hydrodynamic and magnetodynamic effects. An existence theorem of global solutions with large discontinuous initial data is also established.

Key Words: magnetohydrodynamics (MHD), free boundary, global solutions, existence, uniqueness, regularity, large discontinuities.

Mathematics Subject Classification: 35L65, 35B45, 35B40, 76N10, 76W05, 76X05.

1. INTRODUCTION

Magnetohydrodynamics (MHD) concerns the motion of conducting fluids, such as gases, in an electromagnetic field. The applications of magnetohydrodynamics cover 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. If a conducting fluid moves in a magnetic field, electric fields are induced and an electric current flow is developed. The magnetic field exerts forces on these currents which considerably modify the hydrodynamic motion of the fluid. On the other hand, the development of electric currents effects a change in the magnetic field. There is a complex interaction between the magnetic and fluid dynamic phenomena, and both hydrodynamic and electrodynamic effects have to be considered. Plane magnetohydrodynamic flows are governed by the following equations (see Appendix):

ρ_{t} + (ρu)_{x} = 0, x ∈ R, t > 0,

(ρu)_{t} + (ρu^{2} + p + \frac{1}{2}|b|^{2})_{x} = (λu_{x})_{x},

(ρw)_{t} + (ρuw − b)_{x} = (μw_{x})_{x},

b_{t} + (ub − w)_{x} = (νb_{x})_{x},

E_{t} + (u (E + p + \frac{1}{2}|b|^{2}) − w · b)_{x} = (λuu_{x} + μw · w_{x} + νb · b_{x} + κθ_{x})_{x},

(1.1)

where ρ 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 θ the temperature; the longitudinal magnetic field is a constant which is taken to be one in (1.1); the total energy of the plane magnetohydrodynamic flows is

E = ρ (e + \frac{1}{2}(u^{2} + |w|^{2})) + \frac{1}{2}|b|^{2},

with the internal energy e; both the pressure p and the internal energy e are related with the density and temperature of the flow via the equations of state:

p = p(ρ, θ), e = e(ρ, θ);

λ = λ(ρ, θ) and μ = μ(ρ, θ) are the viscosity coefficients of the flow, ν = ν(ρ, θ) is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field, κ = κ(ρ, θ) is the heat conductivity, and all these kinetic coefficients and the magnetic diffusivity are independent of the magnitude and direction of the magnetic field (cf. [23]). The equations in (1.1) describe the macroscopic behavior of the magnetohydrodynamic flow with dissipative mechanisms. This is a three-dimensional magnetohydrodynamic flow which is uniform in the transverse directions.

There have been a lot of studies on magnetohydrodynamics by physicists and mathematicians because of its physical importance, complexity, rich phenomena, and mathematical challenging; see [1, 3, 4, 6, 9, 15, 16, 21, 23, 24, 25, 30, 31] and the references cited therein. The initial-boundary value problems with fixed or free boundaries for this system are important and fundamental. In this paper we consider such a fundamental problem for the magnetohydrodynamic fluid flow with the pressure, internal energy, and heat conductivity satisfying certain physical growth conditions on the temperature. In particular, the case of perfect gases with

p = Rρθ,  e = c_{v}θ,  (1.2)

is included in the class of fluids investigated in this paper, where R is the gas constant, c_{v} = R/(γ − 1) is the heat capacity of the gas at constant volume, and γ is the adiabatic exponent,

We are interested in the well-posedness and regularity of global solutions of the initial-boundary value problem. For perfect gases with small smooth initial data, the existence of global solutions was proved in [15], and the large-time behavior was studied in [25]. For large initial data, these problems have additional difficulties because of the presence of the magnetic field and its interaction with the hydrodynamic motion of the flow of large oscillation.

In this paper we focus on a free boundary problem of (1.1) for x ∈ Ω_{t} = (0, x(t)), t > 0, with the following initial-boundary conditions:

(ρ, u, w, b, θ)|_{t=0} = (ρ_{0}, u_{0}, w_{0}, b_{0}, θ_{0})(x), x ∈ Ω_{0} = (0, 1),

(w, b, θ_{x})|_{∂Ω_{t}} = 0, u|_{x=0} = 0, (p − λu_{x})|_{x=x(t)} = 1, (1.3)

where x = x(t) is the free boundary defined by x′(t) = u(x(t), t) with x(0) = 1, some outer pressure (or stress) (taken to be one without loss of generality) is applied along the free boundary, and the initial values are compatible with the boundary conditions. The main point in our analysis is whether shock waves, vacuum, and concentration are developed in the solutions in a finite time under the outer pressure, provided the initial data is bounded, smooth, and does not contain vacuum. As well-known, for the inviscid case, such phenomena may occur.

To treat the free boundary, we introduce a Lagrangian variable and transform the initial-boundary value problem (1.1) and (1.3) with the free boundary in Euler coordinates into the corresponding initial-boundary value problem with some fixed boundary in Lagrangian coordinates. These two problems are equivalent for the solutions under consideration.

We first establish the existence and uniqueness of global solutions to the initial-boundary value problem with general large initial data in H^{1} (as well as in a Hölder space) and show that neither shock waves nor vacuum and concentration are developed in a finite time for such initial data under outer pressure. There were some similar fundamental results on the nonlinear thermoviscoelasticity [7, 8] and on viscous heat-conductive real gases [17]. Also see [13] and the references cited therein for some recent discussions. We consider the real magnetohydrodynamic flows with general pressure and internal energy, and permit the generation of heat by the magnetic field, as well as its interaction with the fluid motion. The existence and uniqueness of local solutions can be obtained by using the Banach theorem and the contractivity of the operator defined by the linearization of the problem on a small time interval (cf. Nash [26]). The existence of global solutions is proved by extending the local solutions globally in time based on the global a-priori estimates of solutions.

We first obtain an entropy-type energy estimate involving the dissipative effects of viscosity, magnetic diffusion, and heat diffusion, which is essential to deduce the lower and upper bounds of the density. Some new techniques are developed to achieve these bounds. With these bounds, all the required a-priori estimates are obtained subsequently by our careful analysis and techniques with the aid of some arguments and ideas in [7, 8, 17], and the existence of global solutions in H^{1} (as well as in a Hölder space) is established. In particular, the estimates of the first derivatives of u and the temperature θ are complicated because of the complexity of the system for the real flow, and will be achieved by developing some detailed analysis of the energy equation. The boundedness of the temperature will be proved by a maximum principle. For the case of discontinuous initial data, the initial discontinuities propagate as time evolves (see [5] and [12]). We establish the existence of global weak solutions with large discontinuous initial data for the perfect gases (1.2) by mollifying the initial data and using the estimates obtained above for continuous solutions and a convergence argument in [28, 29].

We state our main results in Section 2, prove the existence of global solutions with initial data in H^{1} (and in a Hölder space) in Section 3, and show the existence of global weak solutions with large discontinuous initial data in Section 4. Section 5 is an appendix for the physical background of the magnetohydrodynamic equations and the derivation of the equations in (1.1).

2. MAIN RESULTS

We consider the free boundary problem (1.1) and (1.3) with positive lower and upper bounds of the initial density and temperature: C_{0}^{−1} ≤ ρ_{0} ≤ C_{0}, M_{0}^{−1} ≤ θ_{0} ≤ M_{0}, for some constants C_{0}, M_{0} > 0. Without loss of generality, we assume ∫_{0}^{1} ρ_{0}(x)dx = 1. We first suppose that ρ, θ > 0, then we will prove the positive lower bounds of ρ and θ later. Introduce the Lagrangian variables (y, t) with y = y(x, t) by

y_{x} = ρ, y_{t} = −ρu; y(0, 0) = 0.

Then

y = y(x, t) = ∫_{0}^{x} ρ(ξ, t)dξ,

since u(0, t) = 0. Using the first equation in (1.1) and the free boundary condition, one has y(x(t), t) = y(x(0), 0) = ∫_{0}^{1} ρ_{0}dx = 1. Thus 0 ≤ y ≤ 1. The free boundary problem (1.1) and (1.3) in Euler coordinates (x, t) is transformed into the following problem in Lagrangian coordinates (y, t) for y ∈ Ω = [0, 1], t > 0:

v_{t} − u_{y} = 0,  (2.1a)
u_{t} + (p + \frac{1}{2}|b|^{2})_{y} = (\frac{λu_{y}}{v} )_{y} ,  (2.1b)
w_{t} − b_{y} = (\frac{μw_{y}}{v} )_{y} ,  (2.1c)
(vb)_{t} − w_{y} = (\frac{νb_{y}}{v} )_{y} ,  (2.1d)

E_{t} + (u (p + \frac{1}{2}|b|^{2}) − w · b)_{y} = (\frac{λuu_{y}}{v} + \frac{μw · w_{y}}{v} + \frac{νb · b_{y}}{v} + \frac{κθ_{y}}{v} )_{y} ,

(2.1e)

with the following initial-boundary conditions:

(v, u, w, b, θ)|_{t=0} = (v_{0}, u_{0}, w_{0}, b_{0}, θ_{0})(y), y ∈ Ω,

(w, b, θ_{y})|_{∂Ω} = 0, u|_{y=0} = 0, (p − λv^{−1}u_{y}) |_{y=1} = 1, t > 0, (2.2)

where v = 1/ρ is the specific volume, p = p(v, θ), e = e(v, θ), and

E = e + \frac{1}{2}(u^{2} + |w|^{2} + v|b|^{2}).

The second law of thermodynamics states the relation between p and e:

e_{v}(v, θ) + p(v, θ) = θp_{θ}(v, θ).  (2.3)

We assume that p and e are continuously differentiable and κ is twice continuously differentiable in v > 0 and θ ≥ 0; λ, μ, and ν depend only on v with continuous first derivatives in v > 0 such that λ_{1} ≤ λ ≤ λ_{2}, μ_{1} ≤ μ ≤ μ_{2}, ν_{1} ≤ ν ≤ ν_{2} for v > 0 and for some positive constants λ_{i}, μ_{i}, ν_{i} (i = 1, 2). We also assume the growth conditions with exponents r ∈ [0, 1] and q ≥ 2 + 2r such that

(1). there exists a constant e_{0} > 0 so that, for v > 0 and θ ≥ 0,

p_{v}(v, θ) ≤ 0, e(v, θ) ≥ e_{0}(1 + θ^{1+r});  (2.4)

(2). for any given v_{1} > 0, there exist positive constants κ_{0} = κ_{0}(v_{1}), p_{0} = p_{0}(v_{1}), e_{1} = e_{1}(v_{1}) such that, for v ≥ v_{1}, θ ≥ 0,

0 ≤ p(v, θ) ≤ p_{0} \frac{1 + θ^{1+r}}{v} , κ(v, θ) ≥ κ_{0}(1+θ^{q}), e_{θ}(v, θ) ≥ e_{1}(1+θ^{r}); (2.5)

(3). for any given v_{2} > v_{1} > 0, there exist positive constants p_{i} = p_{i}(v_{1}, v_{2}) (i = 1, 2, 3), e_{j} = e_{j}(v_{1}, v_{2}) (j = 2, 3), and κ_{1} = κ_{1}(v_{1}, v_{2}) so that, for any v ∈ [v_{1}, v_{2}], θ ≥ 0,

|p_{θ}(v, θ)| ≤ p_{1} \frac{1 + θ^{r}}{v} , −p_{3} \frac{1 + θ^{1+r}}{v^{2}} ≤ p_{v}(v, θ) ≤ −p_{2} \frac{1 + θ^{1+r}}{v^{2}} , (2.6)
|e_{v}(v, θ)| ≤ e_{2}(1 + θ^{1+r}), e_{θ}(v, θ) ≤ e_{3}(1 + θ^{r}),  (2.7)
κ(v, θ) + |κ_{v}(v, θ)| + |κ_{vv}(v, θ)| ≤ κ_{1}(1 + θ^{q}).  (2.8)

These growth conditions are motivated by the physical facts: e ∝ θ^{1+r} with r ≈ 0.5 and κ ∝ θ^{5/2} for important physical regimes (see [2, 6, 30, 32]). The perfect gases are included which correspond to the special case r = 0. For the problem (2.1) and (2.2), we will see that, if the initial data is in H^{1} (or a Hölder space), then the solution will be at least in H^{1} (or the Hölder space), and neither shock waves nor vacuum and concentration are developed in a finite time. Precisely, we have the following results.

Theorem 2.1. Suppose that there are two positive constants C_{0}, M_{0} such that

\begin{matrix}C_{0}^{−1} ≤ v_{0}(y) ≤ C_{0}, M_{0}^{−1} ≤ θ_{0}(y) ≤ M_{0}, y ∈ Ω, \\ ∥(u_{0}, w_{0}, b_{0})∥_{L^{4}} + ∥θ_{0}∥_{L^{2}} ≤ C_{0}, ∥(v_{0}, u_{0}, w_{0}, b_{0}, θ_{0})∥_{H^{1}} ≤ M_{0}, \end{matrix}

and v_{0} ∈ W^{1,∞}(Ω). Then the problem (2.1) and (2.2) has a unique global solution (v, u, w, b, θ)(y, t) such that, for any fixed T > 0,

v ∈ L^{∞}(0, T; H^{1} ∩ W^{1,∞}(Ω)), (u, w, b, θ) ∈ L^{∞}(0, T; H^{1}(Ω)),

and, for each (y, t) ∈ Ω × [0, T],

C^{−1} ≤ v(y, t) ≤ C, M^{−1} ≤ θ(y, t) ≤ M, |(u, w, b)(y, t)| ≤ M, ∥(u, w, b)∥_{L^{2}(0,T;L^{4}∩H^{1})} + ∥θ∥_{L^{2}(0,T;L^{2}∩H^{1})} ≤ C,

∥(v, u, w, b, θ)∥_{H^{1}}^{2}(t) + ∫_{0}^{t} ∥(v_{yt}, u_{yy}, w_{yy}, b_{yy}, θ_{yy})∥_{L^{2}}^{2}(s)ds ≤ M,

(2.9)

where C > 0 is some constant depending only on C_{0} and T, and M = M(C_{0}, M_{0}, T) > 0. If v_{0} ∈ C^{1+α}(Ω) and (u_{0}, w_{0}, b_{0}, θ_{0}) ∈ C^{2+α}(Ω) for some α ∈ (0, 1), there exists a unique classical solution v ∈ C^{1+α,1+α/2}(Ω× [0, T]) and (u, w, b, θ) ∈ C^{2+α,1+α/2}(Ω × [0, T]) for any fixed T > 0 to the problem (2.1) and (2.2) satisfying (2.9).

The existence of local solutions is known from the standard method based on the Banach theorem and the contractivity of the operator defined by the linearization of the problem on a small time interval (cf. [26]). The global existence of solutions will be proved by the method of extending the local solutions with respect to time based on a-priori global estimates.

Moreover, we consider the perfect gases in the sense of the state equations (1.2) with constant λ, μ, ν and with κ independent of v satisfying (2.5) and (2.8) for q ≥ 2. Then we have the following theorem for large discontinuous initial data.

Theorem 2.2. Suppose that there exists C_{0} > 0 such that

C_{0}^{−1} ≤ v_{0}(y) ≤ C_{0}, θ_{0}(y) ≥ 0, ∥(u_{0}, w_{0}, b_{0})∥_{L^{4}} + ∥θ_{0}∥_{L^{2}} ≤ C_{0}.

Then there exists a global weak solution (v, u, w, b, θ)(y, t) to the problem (2.1) and (2.2) for perfect gases, satisfying (2.1) in the sense of distributions such that, for any fixed T > 0,

(u, w, b) ∈ L^{2}(0, T; L^{4}), θ ∈ L^{2}(0, T; L^{2}),

with θ(·, t) ⇀ θ_{0} weakly in L^{2} as t → 0. Furthermore, there is a constant C depending on C_{0} and T such that, for each (y, t) ∈ Ω × [0, T],

\begin{matrix}C^{−1} ≤ v(y, t) ≤ C, θ(y, t) ≥ 0, \\ ∥(u, w, b)∥_{L^{2}(0,T;L^{4}∩H^{1})} + ∥θ∥_{L^{2}(0,T;L^{2}∩H^{1})} ≤ C. \end{matrix}  (2.10)

Remark 2.1. The solutions in Theorem 2.2 can be improved to the case where the solution (u, w, b) ∈ C([0, T]; L^{4}(Ω)) ∩ C((0, T]; H^{1}(Ω)) and θ ∈ C^{2}([0, T]; L^{2}(Ω)) ∩ C((0, T]; H^{1}(Ω)).

The results for the problem (2.1) and (2.2) in Theorems 2.1 and 2.2 in Lagrangian coordinates can easily be converted to equivalent statements for the corresponding results for the problem (1.1) and (1.3) in Euler coordinates. In particular, we have the following results.

Theorem 2.3. For the problem (1.1) and (1.3), the following holds: (1). Suppose that there are some constants C_{0}, M_{0} > 0 such that

C_{0}^{−1} ≤ ρ_{0}(x) ≤ C_{0}, M_{0}^{−1} ≤ θ_{0}(x) ≤ M_{0}, x ∈ (0, 1),
∥(u_{0}, w_{0}, b_{0})∥_{L^{4}(0,1)} + ∥θ_{0}∥_{L^{2}(0,1)} ≤ C_{0},
∥(ρ_{0}, u_{0}, w_{0}, b_{0}, θ_{0})∥_{H^{1}(0,1)} ≤ M_{0},

and ρ_{0} ∈ W^{1,∞}(0, 1). Then the problem (1.1) and (1.3) has a global solution (ρ, u, w, b, θ)(x, t) such that, for any fixed T > 0,

ρ ∈ L^{∞}(0, T; H^{1} ∩ W^{1,∞}), (u, w, b, θ) ∈ L^{∞}(0, T; H^{1}),

and, for each (x, t), x ∈ Ω_{t}, t ∈ [0, T],

C^{−1} ≤ ρ(x, t) ≤ C, M^{−1} ≤ θ(x, t) ≤ M, |(u, w, b)(x, t)| ≤ M, ∥(u, w, b)∥_{L^{2}(0,T;L^{4}∩H^{1})} + ∥θ∥_{L^{2}(0,T;L^{2}∩H^{1})} ≤ C,

∥(v, u, w, b, θ)∥_{H^{1}(Ω_{t})}^{2}(t) + ∫_{0}^{t} ∥(v_{xt}, u_{xx}, w_{xx}, b_{xx}, θ_{xx})∥_{L^{2}(Ω_{s})}^{2}(s)ds ≤ M,

(2.11)

where C > 0 is a constant independent of M_{0}, and M = M(C_{0}, M_{0}, T) > 0. If ρ_{0} ∈ C^{1+α}(0, 1) and (u_{0}, w_{0}, b_{0}, θ_{0}) ∈ C^{2+α}(0, 1) for some α ∈ (0, 1), there exists a classical solution ρ ∈ C^{1+α,1+α/2} and (u, w, b, θ) ∈ C^{2+α,1+α/2} for any fixed T > 0 to the problem (1.1) and (1.3) satisfying (2.11). (2). Suppose that there exists C_{0} > 0 such that, for x ∈ (0, 1),

C_{0}^{−1} ≤ ρ_{0}(x) ≤ C_{0}, θ_{0}(x) ≥ 0, ∥(u_{0}, w_{0}, b_{0})∥_{L^{4}(0,1)} + ∥θ_{0}∥_{L^{2}(0,1)} ≤ C_{0}.

Then the problem (1.1) and (1.3) for perfect gases has a global weak solution (ρ, u, w, b, θ)(x, t) satisfying (1.1) and (1.3) in the sense of distributions such that, for any fixed T > 0,

(u, w, b) ∈ L^{2}(0, T; L^{4}), θ ∈ L^{2}(0, T; L^{2}),

with θ(·, t) ⇀ θ_{0} weakly in L^{2} as t → 0. Furthermore, there is a constant C depending on C_{0} and T such that, for each (x, t), x ∈ Ω_{t}, t ∈ [0, T],

\begin{matrix}C^{−1} ≤ ρ(x, t) ≤ C, θ(x, t) ≥ 0, \\ ∥(u, w, b)∥_{L^{2}(0,T;L^{4}∩H^{1})} + ∥θ∥_{L^{2}(0,T;L^{2}∩H^{1})} ≤ C. \end{matrix}  (2.12)

Remark 2.2. If the constant outer pressure in (1.3) is replaced by a general outer pressure R(t) > 0, the results in Theorem 2.3 are still true, and the proof follows directly from the techniques developed here. The case of free outer pressure R(t) = 0 needs new techniques which is handled in a forthcoming paper. For the initial-boundary value problem of (1.1) for x ∈ Ω = (0, 1) and t > 0 with the simple fixed boundary condition: (u, w, b, θ_{x})|_{∂Ω} = 0, similar results can be obtained by using the techniques of this paper with certain modifications.

3. A-PRIORI ESTIMATES

In order to establish the global existence, we need a-priori estimates of the solutions for (y, t) ∈ Ω × [0, T] for any fixed T > 0. We denote C > 0 the generic constant, which may depend on C_{0} and T but independent of M_{0}, and denote M > 0 the generic constant, which may depend on C_{0}, M_{0}, and T.

To obtain the complete global a-priori estimates, we first prove the positive lower and upper bounds of the density and some entropy-type energy estimates, then we estimate the first derivatives of v, w, and b. The estimates on the first derivatives of u and θ are complicated because of the complexity of the system of the real flow, which will be achieved by detailed analysis of the energy equation. The boundedness of the temperature will be proved by a maximum principle.

First, from the equations (2.1) and the initial-boundary conditions (2.2), we have the following entropy-type energy estimates and the lower and upper bounds of the density.

Lemma 3.1.

C^{−1} ≤ v(y, t) ≤ C,  (3.1)
∫_{0}^{1} (|(u, w, \sqrt{v}b)|^{2} + v + θ + θ^{1+r} + (θ − 1 − ln θ) + e) dy ≤ C, (3.2)

∫_{0}^{t}∫_{0}^{1} (\frac{κθ_{y}^{2}}{θ^{2}} + λu_{y}^{2} + μ|w_{y}|^{2} + ν|b_{y}|^{2} + θ_{y}^{2}) dy ds ≤ C.  (3.3)

Proof. We first prove the lower bound of v. Integrating the energy equation (2.1e), and using the initial-boundary condition (2.2), integration by parts, and u(1, t) = ∫_{0}^{1} u_{y}dy = ∫_{0}^{1} v_{t}dy, we have

\begin{matrix}∫_{0}^{1} E(y, t)dy − ∫_{0}^{1} E(y, 0)dy = − ∫_{0}^{t} (∫_{0}^{1} vdy)_{t} ds \\ = − ∫_{0}^{1} v(y, t)dy + ∫_{0}^{1} v_{0}(y)dy, \end{matrix}

and then

∫_{0}^{1} (E + v) (y, t)dy = ∫_{0}^{1} (E(y, 0) + v_{0}(y)) dy ≤ C,

that is,

∫_{0}^{1} (e + |(u, w, \sqrt{v}b)|^{2} + v) dy ≤ C.

From (2.4), ∫_{0}^{1} (θ + θ^{1+r}) dy ≤ C + C ∫_{0}^{1} e dy ≤ C. Set V (v) ≡ ∫_{1}^{v} \frac{λ(ξ)}{ξ}dξ, and then λ_{1} ln v ≤ V (v) ≤ λ_{2} ln v, V_{t} = \frac{λ}{v}u_{y}, and (2.1b) becomes u_{t} = (V_{t} − p − \frac{1}{2}|b|^{2})_{y}. Integrating this equation over [y, 1] × [0, t] yields

V (v(y, t)) = V (v_{0}(y)) + ∫_{0}^{t} (p + \frac{1}{2}|b|^{2}) ds − ∫_{y}^{1}(u(ξ, t) − u_{0}(ξ))dξ − t.

(3.4)

Then V (v(y, t)) ≥ −C for any (y, t) ∈ [0, 1] × [0, T] since p ≥ 0 and

|∫_{y}^{1}(u(ξ, t) − u_{0}(ξ))dξ| ≤ (2 ∫_{0}^{1}(u^{2} + u_{0}^{2})dy)^{1/2} ≤ C,

which implies the positive lower bound of v, i.e. v ≥ C^{−1}.

In order to prove the upper bound of v, we need some entropy-type energy estimates. Define η(v, θ) by the relations: e_{θ} = θη_{θ}, η_{v} = p_{θ}. Set ψ(v, θ) = e(v, θ) − θη(v, θ). Then e_{θ} = −θψ_{θθ} and ψ_{v} = −p. Set

W (v, θ) = ψ(v, θ) − ψ(v, 1) − ψ_{θ}(v, θ)(θ − 1).

Then

(W + \frac{1}{2}(u^{2} + |w|^{2} + v|b|^{2}))_{t} + \frac{κθ_{y}^{2}}{vθ^{2}} + \frac{λu_{y}^{2}}{vθ} + \frac{μ|w_{y}|^{2}}{vθ} + \frac{ν|b_{y}|^{2}}{vθ}

= p(v, 1)v_{t} + (\frac{λuu_{y}}{v} + \frac{μw · w_{y}}{v} + \frac{νb · b_{y}}{v} + \frac{κθ_{y}}{v} )_{y}

− (u (p + \frac{1}{2}|b|^{2}) − w · b + \frac{κθ_{y}}{vθ} )_{y} ,

where p(v, 1) is the value of p(v, θ) at θ = 1. Integrating the above equation over [0, 1] × [0, t] and using (2.5) yield

∫_{0}^{1} (W + \frac{1}{2}(u^{2} + |w|^{2} + v|b|^{2})) dy

+ ∫_{0}^{t}∫_{0}^{1} (\frac{κθ_{y}^{2}}{vθ^{2}} + \frac{λu_{y}^{2}}{vθ} + \frac{μ|w_{y}|^{2}}{vθ} + \frac{ν|b_{y}|^{2}}{vθ} ) dy ds

≤ C + C ∫_{0}^{t}∫_{0}^{1} \frac{|u_{y}|}{v} dy ds ≤ C + \frac{1}{2} ∫_{0}^{t}∫_{0}^{1} \frac{λu_{y}^{2}}{vθ} dy ds + C ∫_{0}^{t}∫_{0}^{1} θ dy ds.

Then

\begin{matrix}∫_{0}^{1} (W + (u^{2} + |w|^{2} + v|b|^{2})) dy \\ + ∫_{0}^{t}∫_{0}^{1} (\frac{κθ_{y}^{2}}{vθ^{2}} + \frac{λu_{y}^{2}}{vθ} + \frac{μ|w_{y}|^{2}}{vθ} + \frac{ν|b_{y}|^{2}}{vθ} ) dy ds ≤ C. \end{matrix}

By (2.5), one has

W (v, θ) = ψ(v, θ) − ψ(v, 1) − ψ_{θ}(v, θ)(θ − 1)

≥ (θ − 1)^{2} ∫_{0}^{1} \frac{e_{θ}(v, 1 + s(θ − 1))}{1 + s(θ − 1)} s ds

≥ \frac{1}{2}e_{1}(θ − 1 − ln θ) − C_{1}(θ + θ^{1+r}) − C_{2},

where C_{1} and C_{2} are some positive constants. Then

∫_{0}^{1} (θ − 1 − ln θ) dy + ∫_{0}^{t}∫_{0}^{1} (\frac{κθ_{y}^{2}}{vθ^{2}} + \frac{λu_{y}^{2}}{vθ} + \frac{μ|w_{y}|^{2}}{vθ} + \frac{ν|b_{y}|^{2}}{vθ} ) dy ds ≤ C.

Now we estimate the upper bound of v. For any t ∈ [0, T], there exists y(t) ∈ [0, 1] such that θ̄ = θ(y(t), t) = ∫_{0}^{1} θdy ≤ C. From (2.5) and the estimates above, we have

\begin{matrix}∫_{0}^{t} pds ≤ & C ∫_{0}^{t}(1 + θ^{1+r})ds ≤ C ∫_{0}^{t}(1 + θ^{2})ds \\ ≤ & C + C ∫_{0}^{t} (θ̄ + ∫_{y(s)}^{y} θ_{y}(ξ, s)dξ)^{2} ds \\ ≤ & C + C ∫_{0}^{t}∫_{0}^{1} \frac{θ_{y}^{2}}{v} dy ∫_{0}^{1} v dy ds ≤ C + C ∫_{0}^{t}∫_{0}^{1} \frac{κθ_{y}^{2}}{vθ^{2}} dy ds ≤ C. \end{matrix}

From (2.1), we have

e_{t} + pu_{y} = (\frac{κθ_{y}}{v} )_{y} + \frac{λu_{y}^{2}}{v} + \frac{μ|w_{y}|^{2}}{v} + \frac{ν|b_{y}|^{2}}{v} .  (3.5)

Integrating the equation (3.5) and using (2.5), we obtain

\begin{matrix}∫_{0}^{t}∫_{0}^{1} (\frac{λu_{y}^{2}}{v} + \frac{μ|w_{y}|^{2}}{v} + \frac{ν|b_{y}|^{2}}{v} ) dy ds \\ ≤ C + \frac{1}{2} ∫_{0}^{t}∫_{0}^{1} \frac{λu_{y}^{2}}{v} dy ds + C ∫_{0}^{t}∫_{0}^{1} \frac{1 + θ^{2+2r}}{v} dy ds, \end{matrix}

and then

∫_{0}^{t}∫_{0}^{1} (\frac{λu_{y}^{2}}{v} + \frac{μ|w_{y}|^{2}}{v} + \frac{ν|b_{y}|^{2}}{v} ) dy ds

≤ C + C ∫_{0}^{t}∫_{0}^{1} (θ̄^{1+r} + ∫_{0}^{1} θ^{r}|θ_{y}|dy)^{2} dy ds

≤ C + C ∫_{0}^{t}∫_{0}^{1} \frac{κθ_{y}^{2}}{vθ^{2}} dy ∫_{0}^{1} v dy ds ≤ C.

Thus, from the boundary condition (2.2) on b and the estimates above,

\begin{matrix}∫_{0}^{t} \frac{1}{2}|b|^{2}ds = ∫_{0}^{t} ∫_{0}^{y} b · b_{y}ds \\ ≤ ∫_{0}^{t}∫_{0}^{1} v|b|^{2} dy ds + C ∫_{0}^{t}∫_{0}^{1} \frac{ν|b_{y}|^{2}}{v} dy ds ≤ C. \end{matrix}

Therefore, V (v(y, t)) ≤ C, which implies the upper bound of v, that is, v ≤ C. The proof of Lemma 3.1 is completed.

We now make estimates on the first derivatives of v, w, and b.

Lemma 3.2.

\begin{matrix}∫_{0}^{1} (v_{y}^{2} + |w_{y}|^{2} + |b_{y}|^{2}) dy \\ + ∫_{0}^{t}∫_{0}^{1} (|w_{yy}|^{2} + |b_{yy}|^{2} + u_{y}^{4} + |w_{y}|^{4} + |b_{y}|^{4}) dy ds ≤ M. \end{matrix} (3.6)

Proof. Set h(y, t) = ∫_{y}^{1} u(ξ, t)dξ. Equation (2.1b) yields

h_{t} − \frac{λ}{v}h_{yy} = p + \frac{1}{2}|b|^{2} − 1

with the following initial-boundary conditions:

h(y, 0) = ∫_{y}^{1} u_{0}(ξ)dξ, h(0, t) = ∫_{0}^{1} u(y, t)dy, h(1, t) = 0. Then, from (2.5), we have the following estimate (see [8, 22]):

\begin{aligned}∫_{0}^{t}∫_{0}^{1} u_{y}^{4} dy ds = ∫_{0}^{t}∫_{0}^{1} h_{yy}^{4} dy ds & ≤ M + M ∫_{0}^{t}∫_{0}^{1} (p^{4} + |b|^{8}) dy ds \\ & ≤ M + M ∫_{0}^{t}∫_{0}^{1} (|b|^{8} + θ^{4+4r}) dy ds. \end{aligned}

Rewrite (2.1b) as:

(V_{y} − u)_{t} = (p + \frac{1}{2}|b|^{2})_{y} .

Multiplying the above equation by V_{y} − u and integrating it, we have

\frac{1}{2} ∫_{0}^{1} |V_{y} − u|^{2} dy ≤ M + ∫_{0}^{t}∫_{0}^{1} (p_{v}v_{y} + p_{θ}θ_{y} + b · b_{y}) (V_{y} − u) dy ds.

Using (2.6), V_{y} = λv_{y}/v, and the following observation

∫_{0}^{t} max_{y∈Ω} (1 + θ^{1+r}) ds + ∫_{0}^{t} max_{y∈Ω} (1 + θ^{2+2r}) ds + ∫_{0}^{t} max_{y∈Ω} (1 + θ^{1+r})^{2} ds

≤ M ∫_{0}^{t} max_{y∈Ω} (1 + θ^{1+r})^{2} ds ≤ M ∫_{0}^{t} (1 + θ̄^{1+r} + (1 + r) ∫_{0}^{1} θ^{r}|θ_{y}|dy) ds

≤ M + M ∫_{0}^{t}∫_{0}^{1} \frac{κθ_{y}^{2}}{θ^{2}} dy ds ≤ M,

(3.7) we obtain that there exists some constant M_{1} > 0 such that

\frac{1}{2} ∫_{0}^{1} |V_{y} − u|^{2} dy

≤ M − M_{1} ∫_{0}^{t}∫_{0}^{1} (1 + θ^{1+r}) v_{y}^{2} dy ds + M ∫_{0}^{t}∫_{0}^{1} (1 + θ^{1+r}) |v_{y}u| dy ds

+ M ∫_{0}^{t}∫_{0}^{1} (1 + θ^{r}) |θ_{y}| (|V_{y}| + |u|) dy ds + ∫_{0}^{t}∫_{0}^{1} b · b_{y} (V_{y} − u) dy ds

≤ M − \frac{M_{1}}{2} ∫_{0}^{t}∫_{0}^{1} (1 + θ^{1+r}) v_{y}^{2} dy ds + M ∫_{0}^{t} max_{y∈Ω} (1 + θ^{1+r}) ∫_{0}^{1} u^{2} dy ds

+ M ∫_{0}^{t}∫_{0}^{1} \frac{κθ_{y}^{2}}{θ^{2}} dy ds + M ∫_{0}^{t}∫_{0}^{1} |b · b_{y}|^{2} dy ds + M ∫_{0}^{t}∫_{0}^{1} u^{2} dy ds

≤ M − \frac{M_{1}}{2} ∫_{0}^{t}∫_{0}^{1} (1 + θ^{1+r}) v_{y}^{2} dy ds + M ∫_{0}^{t}∫_{0}^{1} |b · b_{y}|^{2} dy ds.

Then ∫_{0}^{1} |V_{y} − u|^{2} dy + ∫_{0}^{t}∫_{0}^{1} (1 + θ^{1+r}) v_{y}^{2} dy ds ≤ M + M ∫_{0}^{t}∫_{0}^{1} |b · b_{y}|^{2} dy ds, and

\begin{aligned}∫_{0}^{1} v_{y}^{2}dy & ≤ M ∫_{0}^{1} V_{y}^{2}dy ≤ M ∫ |V_{y} − u|^{2} dy + M ∫_{0}^{1} u^{2}dy \\ & ≤ M + M ∫_{0}^{t}∫_{0}^{1} |b · b_{y}|^{2} dy ds. \end{aligned}  (3.8)

Multiplying (2.1e) by K_{1}E and (2.1b) by K_{2}u^{3}, taking the inner product of (2.1c) with K_{3}|w|^{2}w and (2.1d) with K_{4}|vb|^{2}vb, multiplying (3.8) by K_{5}, respectively, for proper positive constants K_{j}, 1 ≤ j ≤ 5, integrating them over [0, 1]×[0, t], adding them all together, and using Gronwall’s inequality, we have ∫_{0}^{1} v_{y}^{2}dy + ∫_{0}^{t}∫_{0}^{1} |b · b_{y}|^{2} dy ds ≤ M by tedious calculations. Using Young’s inequality, we have ∫_{0}^{t}∫_{0}^{1} (|b|^{8} + θ^{4+4r}) dy ds

≤ ∫_{0}^{t} max_{y∈Ω} |b|^{6} ∫_{0}^{1} |b|^{2} dy ds + ∫_{0}^{t} max_{y∈Ω} θ^{3+3r} ∫_{0}^{1} θ^{1+r} dy ds

≤ M ∫_{0}^{t} (θ̄^{3+3r} + ∫_{0}^{1} |b|^{6}dy + ∫_{0}^{1} |b|^{4}|b · b_{y}| + ∫_{0}^{1} θ^{2+2r} · θ^{r}|θ_{y}|dy) ds

≤ \frac{1}{2} ∫_{0}^{t}∫_{0}^{1} |b|^{8} dy ds + \frac{1}{2} ∫_{0}^{t}∫_{0}^{1} θ^{4+4r} dy ds

+ M (1 + ∫_{0}^{t}∫_{0}^{1} |b · b_{y}|^{2} dy ds + ∫_{0}^{t}∫_{0}^{1} \frac{κθ_{y}^{2}}{θ^{2}} dy ds)

≤ M + \frac{1}{2} ∫_{0}^{t}∫_{0}^{1} |b|^{8} dy ds + \frac{1}{2} ∫_{0}^{t}∫_{0}^{1} θ^{4+4r} dy ds,

and then

∫_{0}^{t}∫_{0}^{1} (|b|^{8} + θ^{4+4r}) dy ds ≤ M,

and thus ∫_{0}^{t}∫_{0}^{1} u_{y}^{4} dy ds ≤ M. Multiplying (2.1c) by w_{yy} and then integrating, and using the interpolation inequality:

|w_{y}|^{2} ≤ (1 + δ^{−1}) ∫_{0}^{1} |w_{y}|^{2}dy + δ ∫_{0}^{1} |w_{yy}|^{2}dy, for any δ > 0, we have

\frac{1}{2} ∫_{0}^{1} |w_{y}|^{2}dy = \frac{1}{2} ∫_{0}^{1} |w_{y}(y, 0)|^{2}dy − ∫_{0}^{t}∫_{0}^{1} (b + \frac{μw_{y}}{v} )_{y} · w_{yy} dy ds

≤ M − M_{2} ∫_{0}^{t}∫_{0}^{1} |w_{yy}|^{2} dy ds + M ∫_{0}^{t}∫_{0}^{1} (|b_{y}| + |v_{y}||w_{y}|) |w_{yy}| dy ds

≤ M − \frac{3M_{2}}{4} ∫_{0}^{t}∫_{0}^{1} |w_{yy}|^{2} dy ds + M ∫_{0}^{t}∫_{0}^{1} |b_{y}|^{2} dy ds

+ M ∫_{0}^{t} max_{y∈Ω} |w_{y}|^{2} ∫_{0}^{1} v_{y}^{2} dy ds

≤ M − \frac{M_{2}}{2} ∫_{0}^{t}∫_{0}^{1} |w_{yy}|^{2} dy ds + M ∫_{0}^{t}∫_{0}^{1} |w_{y}|^{2} dy ds

≤ M − \frac{M_{2}}{2} ∫_{0}^{t}∫_{0}^{1} |w_{yy}|^{2} dy ds,

and then

∫_{0}^{1} |w_{y}|^{2}dy + ∫_{0}^{t}∫_{0}^{1} |w_{yy}|^{2} dy ds ≤ M.

As a consequence,

∫_{0}^{t}∫_{0}^{1} |w_{y}|^{4} dy ds ≤ ∫_{0}^{t} max_{y∈Ω} |w_{y}|^{2} ∫_{0}^{1} |w_{y}|^{2} dy ds

≤ M ∫_{0}^{t}∫_{0}^{1} (|w_{y}|^{2} + |w_{yy}|^{2}) dy ds ≤ M.

Using v_{t} = u_{y}, we rewrite (2.1d) into:

b_{t} = \frac{u_{y}}{v} b + \frac{1}{v} (w + \frac{νb_{y}}{v} )_{y} .  (3.9)

Multiplying the above equation by b_{yy}, integrating it, and using Lemma 3.1 and the interpolation inequalities: for any δ > 0,

\begin{matrix}|b|^{2} ≤ (1 + δ^{−1}) ∫_{0}^{1} |b|^{2}dy + δ ∫_{0}^{1} |b_{y}|^{2}dy, \\ |b_{y}|^{2} ≤ (1 + δ^{−1}) ∫_{0}^{1} |b_{y}|^{2}dy + δ ∫_{0}^{1} |b_{yy}|^{2}dy \end{matrix}

yield

\frac{1}{2} ∫_{0}^{1} |b_{y}|^{2}dy

≤ M − M_{3} ∫_{0}^{t}∫_{0}^{1} |b_{yy}|^{2} dy ds

+ M ∫_{0}^{t}∫_{0}^{1} (|u_{y}||b| + |w_{y}| + |v_{y}||b_{y}|) |b_{yy}| dy ds

≤ M − \frac{3M_{3}}{4} ∫_{0}^{t}∫_{0}^{1} |b_{yy}|^{2} dy ds + M max_{y,t} |b|^{2} ∫_{0}^{t}∫_{0}^{1} u_{y}^{2} dy ds

+ M ∫_{0}^{t} max_{y∈Ω} |b_{y}|^{2} ∫_{0}^{1} v_{y}^{2} dy ds

≤ M − \frac{M_{3}}{2} ∫_{0}^{t}∫_{0}^{1} |b_{yy}|^{2} dy ds + M ∫_{0}^{1} |b|^{2}dy + \frac{1}{4} ∫_{0}^{1} |b_{y}|^{2}dy

+ M ∫_{0}^{t}∫_{0}^{1} |b_{y}|^{2} dy ds

≤ M − \frac{M_{3}}{2} ∫_{0}^{t}∫_{0}^{1} |b_{yy}|^{2} dy ds + \frac{1}{4} ∫_{0}^{1} |b_{y}|^{2}dy.

Then

∫_{0}^{1} |b_{y}|^{2}dy + ∫_{0}^{t}∫_{0}^{1} |b_{yy}|^{2} dy ds ≤ M.

Thus

∫_{0}^{t}∫_{0}^{1} |b_{y}|^{4} dy ds ≤ ∫_{0}^{t} max_{y∈Ω} |b_{y}|^{2} ∫_{0}^{1} |b_{y}|^{2} dy ds
≤ M ∫_{0}^{t}∫_{0}^{1} (|b_{y}|^{2} + |b_{yy}|^{2}) dy ds ≤ M.

The proof of Lemma 3.2 is completed.

We now make estimates on the first derivatives of u and θ.

Lemma 3.3.

∫_{0}^{1} (|u_{y}|^{2} + |u_{yy}|^{2} + θ_{y}^{2}) dy + ∫_{0}^{t}∫_{0}^{1} (θ_{t}^{2} + |b_{t}|^{2}) dy ds ≤ M. (3.10)

Proof. Due to the complicated structures of the pressure, internal energy, and heat conductivity, as well as the strong coupling of the equations, the estimates of u_{y} and θ_{y} are quite complex. Set

Θ = max_{y∈Ω,t∈[0,T]} θ(y, t), X = ∫_{0}^{t}∫_{0}^{1} (1 + θ^{q+r}) θ_{t}^{2} dy ds,

Y = max_{t∈[0,T]} ∫_{0}^{1} (1 + θ^{2q}) θ_{y}^{2}dy, Z = max_{t∈[0,T]} ∫_{0}^{1} u_{yy}^{2}dy.

We will show that Θ, X, and Y can be controlled by Z and then we derive an inequality on Z which yields the upper bound of Z and thus the upper bounds of Θ, X, and Y. The complete proof can be divided into four steps. Step 1. Using the following interpolation inequality (cf. [10])

∫_{0}^{1} u_{y}^{2}dy ≤ M ∫_{0}^{1} u^{2}dy + M (∫_{0}^{1} u^{2}dy)^{1/2} (∫_{0}^{1} u_{yy}^{2}dy)^{1/2} ,

one has

∫_{0}^{1} u_{y}^{2}dy ≤ M + MZ^{1/2},  (3.11)

and

u_{y}^{2} ≤ ∫_{0}^{1} u_{y}^{2}dy + 2 (∫_{0}^{1} u_{y}^{2}dy)^{1/2} (∫_{0}^{1} u_{yy}^{2}dy)^{1/2} ≤ M + MZ^{1/2} + MZ^{3/4},

which implies

|u_{y}| ≤ M + MZ^{3/8}.  (3.12)

From Lemma 3.1, we have

\begin{matrix}θ^{(2q+3+r)/2}(y, t) = θ̄^{(2q+3+r)/2} + \frac{2q + 3 + r}{2} ∫_{y(t)}^{y} θ^{(2q+1+r)/2}θ_{y}dξ \\ ≤ M + M (∫_{0}^{1} θ^{2q}θ_{y}^{2}dy)^{1/2} (∫_{0}^{1} θ^{1+r}dy)^{1/2} ≤ M + MY^{1/2}, \end{matrix}

and thus

Θ ≤ M + MY^{β_{1}},  (3.13)

where β_{1} = 1/(2q + 3 + r). Step 2. We now show that X and Y can be controlled by Z. Use (2.3) and rewrite the equation (3.5) as:

e_{θ}θ_{t} + θp_{θ}u_{y} = (\frac{κθ_{y}}{v} )_{y} + \frac{λu_{y}^{2}}{v} + \frac{μ|w_{y}|^{2}}{v} + \frac{ν|b_{y}|^{2}}{v} .  (3.14)

Set H(v, θ) = v^{−1} ∫_{0}^{θ} κ(v, ξ) dξ. Multiplying (3.14) by H_{t}, using integration by parts and the boundary condition (2.2), we have

\begin{matrix}∫_{0}^{t}∫_{0}^{1} (e_{θ}θ_{t} + θp_{θ}u_{y} − \frac{λ}{v}u_{y}^{2} − \frac{μ}{v}|w_{y}|^{2} − \frac{ν}{v} |b_{y}|^{2}) H_{t} dy ds \\ + ∫_{0}^{t}∫_{0}^{1} \frac{κ}{v}θ_{y}H_{ty} dy ds = 0, \end{matrix}  (3.15)

where

H_{t} = H_{v}u_{y} + \frac{κ}{v}θ_{t}, H_{ty} = H_{v}u_{yy} + H_{vv}u_{y}v_{y} + (\frac{κ}{v})_{v} θ_{t}v_{y} + (\frac{κ}{v}θ_{y})_{t} .

We now estimate all the terms in the above equality. First we have, from (2.4) and (2.5),

∫_{0}^{t}∫_{0}^{1} e_{θ}θ_{t} \frac{κ}{v}θ_{t} dy ds ≥ M_{4}X, ∫_{0}^{T}∫_{0}^{1} \frac{κ}{v}θ_{y} (\frac{κ}{v}θ_{y})_{t} dy ds ≥ M_{5}Y − M,

for some positive constants M_{4} and M_{5}. From (2.8),

|H_{v}| + |H_{vv}| ≤ M ∫_{0}^{θ} (|κ| + |κ_{v}| + |κ_{vv}|) dξ ≤ M (1 + θ^{q+1}) .

Using (2.6), (2.7), (3.12), (3.13), Lemmas 3.1 and 3.2, and Young’s inequality, we have

∫_{0}^{t}∫_{0}^{1} e_{θ}θ_{t}H_{v}u_{y} dy ds ≤ M ∫_{0}^{t}∫_{0}^{1} (1 + θ^{q+r+1}) |θ_{t}||u_{y}| dy ds

≤ \frac{M_{4}}{8} X + M (1 + Θ^{q+2+r}) ∫_{0}^{t}∫_{0}^{1} u_{y}^{2} dy ds

≤ \frac{M_{4}}{8} X + MY^{(q+2+r)β_{1}} + M ≤ \frac{M_{4}}{8} X + \frac{M_{5}}{8} Y + M,

and

∫_{0}^{t}∫_{0}^{1} (θp_{θ}u_{y} − \frac{λ}{v}u_{y}^{2} − \frac{μ}{v}|w_{y}|^{2} − \frac{ν}{v} |b_{y}|^{2}) H_{v}u_{y} dy ds

≤ M (1 + Θ^{q+1}) max_{y,t} |u_{y}|×

∫_{0}^{t}∫_{0}^{1} ((1 + θ^{1+r}) |u_{y}| + u_{y}^{2} + |w_{y}|^{2} + |b_{y}|^{2}) dy ds ≤ M (1 + Θ^{q+2+r}) (1 + Z^{3/8}) ∫_{0}^{t}∫_{0}^{1} (|u_{y}| + u_{y}^{2} + |w_{y}|^{2} + |b_{y}|^{2}) dy ds

≤ M + \frac{M_{5}}{8} Y + MZ^{β_{2}}

with β_{2} = 3(2q + 3 + r)/(8q + 8), and

∫_{0}^{t}∫_{0}^{1} (θp_{θ}u_{y} − \frac{λ}{v}u_{y}^{2} − \frac{μ}{v}|w_{y}|^{2} − \frac{ν}{v} |b_{y}|^{2}) \frac{κ}{v}θ_{t} dy ds

≤ M ∫_{0}^{t}∫_{0}^{1} ((1 + θ^{1+r}) |u_{y}| + u_{y}^{2} + |w_{y}|^{2} + |b_{y}|^{2}) (1 + θ^{q}) |θ_{t}| dy ds

≤ \frac{M_{4}}{8} X + M (1 + Θ^{q+2+r}) ∫_{0}^{t}∫_{0}^{1} u_{y}^{2} dy ds

+ M (1 + Θ^{q−r}) ∫_{0}^{t}∫_{0}^{1} (u_{y}^{4} + |w_{y}|^{4} + |b_{y}|^{4}) dy ds

≤ \frac{M_{4}}{8} X + MΘ^{q+2+r} + M ≤ \frac{M_{4}}{8} X + \frac{M_{5}}{8} Y + M.

Since θ^{q}(y, t) = θ̄^{q} + ∫_{y(t)}^{y} qθ^{q−1}θ_{y}dξ, we use Lemma 3.1 and Young’s inequality to obtain

∫_{0}^{t} max_{y∈Ω} θ^{q}ds ≤ M + M ∫_{0}^{t}∫_{0}^{1} θ^{q−1}|θ_{y}| dy ds

≤ M + ∫_{0}^{t} max_{y∈Ω} θ^{q−1} ∫_{0}^{1} θ dy ds + M ∫_{0}^{t}∫_{0}^{1} \frac{κθ_{y}^{2}}{θ^{2}} dy ds

≤ M + ∫_{0}^{t} max_{y∈Ω} θ^{q−1}ds ≤ M + \frac{1}{2} ∫_{0}^{t} max_{y∈Ω} θ^{q}ds,

and then

∫_{0}^{t} max_{y∈Ω} θ^{q}ds ≤ M.  (3.16)

From (2.4)–(2.8), (3.12), (3.13), (3.16), Lemmas 3.1 and 3.2, and Young’s inequality, we have

∫_{0}^{t}∫_{0}^{1} \frac{κ}{v}θ_{y}H_{vv}u_{y}v_{y} dy ds ≤ M (1 + Θ^{q+1}) max_{y,t} |u_{y}| ∫_{0}^{t}∫_{0}^{1} κ|θ_{y}||v_{y}| dy ds

≤ M (1 + Θ^{q+2}) max_{y,t} |u_{y}| (∫_{0}^{t} max_{y} (1 + θ^{q}) ∫_{0}^{1} v_{y}^{2} dy ds)^{1/2} ×

(∫_{0}^{t}∫_{0}^{1} \frac{κθ_{y}^{2}}{θ^{2}} dy ds)^{1/2}

≤ M (1 + Z^{3/8}) (1 + Y^{(q+2)β_{1}}) ≤ \frac{M_{6}}{8} Y + MZ^{β_{2}} + M.

Using (2.8), (3.13), (3.14), Lemma 3.2, and Young’s inequality yields

∫_{0}^{t}∫_{0}^{1} \frac{κ}{v}θ_{y} (\frac{κ}{v})_{v} θ_{t}v_{y} dy ds ≤ M ∫_{0}^{t}∫_{0}^{1} (1 + θ^{q}) |θ_{t}| |\frac{κ}{v}θ_{y}| |v_{y}| dy ds

≤ \frac{M_{5}}{16} X + M (1 + Θ^{q−r}) ∫_{0}^{t} max_{y} |\frac{κ}{v}θ_{y}|^{2} ∫_{0}^{1} v_{y}^{2} dy ds

≤ \frac{M_{5}}{16} X + M (1 + Θ^{q−r}) ×

× ∫_{0}^{t} (∫_{0}^{1} |\frac{κ}{v}θ_{y}|^{2} dy + 2 ∫_{0}^{1} |\frac{κ}{v}θ_{y}| |(\frac{κ}{v}θ_{y})_{y}| dy) ds

≤ \frac{M_{5}}{16} X + M (1 + Θ^{2q+2−r}) ∫_{0}^{t}∫_{0}^{1} \frac{κθ_{y}^{2}}{θ^{2}} dy ds

+ M (1 + Θ^{q−r}) ∫_{0}^{t}∫_{0}^{1} |\frac{κ}{v}θ_{y}| |(\frac{κ}{v}θ_{y})_{y}| dy ds

≤ \frac{M_{5}}{16} X + M + MY^{(2q+2−r)β_{1}}

+ M (1 + Θ^{q+1−r}) (∫_{0}^{t}∫_{0}^{1} κ (\frac{κ}{v}θ_{y})_{y}^{2} dy ds)^{1/2} (∫_{0}^{t}∫_{0}^{1} \frac{κθ_{y}^{2}}{θ^{2}} dy ds)^{1/2}

≤ \frac{M_{5}}{16} X + \frac{M_{6}}{16} Y + M + M (1 + Θ^{q+1−r}) ×

× (∫_{0}^{t}∫_{0}^{1} κ (e_{θ}^{2}θ_{t}^{2} + θ^{2}p_{θ}^{2}u_{y}^{2} + u_{y}^{4} + |w_{y}|^{4} + |b_{y}|^{4}) dy ds)^{1/2}

≤ \frac{M_{5}}{16} X + \frac{M_{6}}{16} Y + M + M (1 + Θ^{(2q+2−r)/2}) X^{1/2}

+ M (1 + Θ^{(3q+4)/2}) (∫_{0}^{t}∫_{0}^{1} u_{y}^{2} dy ds)^{1/2}

+ M (1 + Θ^{(3q+2−2r)/2}) (∫_{0}^{t}∫_{0}^{1} (u_{y}^{4} + |w_{y}|^{4} + |b_{y}|^{4}) dy ds)^{1/2}

≤ \frac{M_{5}}{16} X + \frac{M_{6}}{16} Y + M + M (1 + Θ^{(2q+2−r)/2}) X^{1/2} + MΘ^{(3q+4)/2} + M

≤ \frac{M_{5}}{8} X + \frac{M_{6}}{16} Y + MΘ^{2q+2−r} + M ≤ \frac{M_{5}}{8} X + \frac{M_{6}}{8} Y + M.

From (2.4), (2.8), (3.12), (3.13), (3.16), Lemma 3.1, and Young’s inequality, we have

∫_{0}^{t}∫_{0}^{1} \frac{κ}{v}θ_{y}H_{v}u_{yy} dy ds ≤ M ∫_{0}^{t}∫_{0}^{1} (1 + θ^{q+1}) κ|θ_{y}||u_{yy}| dy ds

≤ M (1 + Θ^{(4q+5)/4}) (∫_{0}^{t}∫_{0}^{1} \frac{κθ_{y}^{2}}{θ^{2}} dy ds)^{1/2} ×

× (Z ∫_{0}^{t} (1 + max_{y} θ^{(2q+3)/2}) ds)^{1/2}

≤ M (1 + Y^{β_{1}(4q+5)/4}) Z^{1/2} ≤ \frac{M_{6}}{8} Y + MZ^{β_{3}} + M,

where β_{3} = 2(2q + 3 + r)/(4q + 7 + 4r) ∈ (0, 1) and the following estimate is employed:

∫_{0}^{t} max_{y} θ^{(2q+3)/2}ds ≤ M,  (3.17)

which will be shown in the next step. Assuming the estimate (3.17), we conclude, from (3.15), that

X + Y ≤ M + MZ^{β_{4}},  (3.18)

where 0 < β_{4} = max{β_{2}, β_{3}} < 1. Step 3. The proof of (3.17) is as follows. For some small δ, from Lemma 3.1,

∫_{0}^{t} max_{y} θ^{(2q+3)/2}ds ≤ ∫_{0}^{t} (θ̄^{(2q+3)/2} + ∫_{0}^{1} θ^{(2q+1)/2}|θ_{y}|dy) ds

≤ M + δ ∫_{0}^{t} max_{y} θ^{(2q+3)/2} ∫_{0}^{1} θ^{1+r} dy ds + M ∫_{0}^{t}∫_{0}^{1} θ^{(2q−3−2r)/2}θ_{y}^{2} dy ds

≤ M + \frac{1}{2} ∫_{0}^{t} max_{y} θ^{(2q+3)/2}ds + M ∫_{0}^{t}∫_{0}^{1} θ^{(2q−3−2r)/2}θ_{y}^{2} dy ds,

and then

∫_{0}^{t} max_{y} θ^{(2q+3)/2}ds ≤ M + M ∫_{0}^{t}∫_{0}^{1} θ^{(2q−3−2r)/2}θ_{y}^{2} dy ds.

For r = 1, using Lemma 3.1, we obtain (3.17) since

\begin{aligned}∫_{0}^{t} max_{y} θ^{(2q+3)/2}ds & ≤ M + M ∫_{0}^{t}∫_{0}^{1} θ^{(2q−5)/2}θ_{y}^{2} dy ds \\ & ≤ M + M ∫_{0}^{t}∫_{0}^{1} \frac{κθ_{y}^{2}}{θ^{2}} dy ds ≤ M. \end{aligned}

For 0 < r < 1, define G(v, θ) = ∫_{0}^{θ} ξ^{−r}e_{θ}(v, ξ)dξ. Then, from (2.4)–(2.8), Lemma 3.1, and (3.5),

∫_{0}^{1} G(v, θ)dy ≤ M ∫_{0}^{1} (θ + \frac{1}{1 − r} θ^{1−r}) dy ≤ M ∫_{0}^{1}(θ + 1)dy ≤ M,

and

G_{t} + (1 − r)u_{y} ∫_{0}^{θ} ξ^{−r}p_{θ}(v, ξ)dξ = \frac{λu_{y}^{2}}{vθ^{r}} + \frac{rκθ_{y}^{2}}{vθ^{r+1}} + (\frac{κθ_{y}}{vθ^{r}} )_{y} .

Integrating the above equation and using (2.4)–(2.8) and Lemma 3.1, we have

∫_{0}^{t}∫_{0}^{1} (\frac{λu_{y}^{2}}{vθ^{r}} + \frac{rκθ_{y}^{2}}{vθ^{r+1}} ) dy ds

≤ M + M ∫_{0}^{t}∫_{0}^{1} u_{y}^{2} dy ds + M ∫_{0}^{t}∫_{0}^{1}(1 + θ)^{2} dy ds

≤ M + M ∫_{0}^{t}∫_{0}^{1} \frac{κθ_{y}^{2}}{θ^{2}} dy ds ≤ M.

Then, from the above estimate, we have

\begin{aligned}∫_{0}^{t} max_{y} θ^{(2q+3)/2}ds & ≤ M + M ∫_{0}^{t}∫_{0}^{1} θ^{(2q−3−2r)/2}θ_{y}^{2} dy ds \\ & ≤ M + M ∫_{0}^{t}∫_{0}^{1} \frac{κθ_{y}^{2}}{θ^{1+r}} dy ds ≤ M. \end{aligned}

If r = 0, replace the above definition of G by G(v, θ) = ∫_{0}^{θ} ξ^{−1/2}e_{θ}(v, ξ)dξ and follow the same procedure to obtain

∫_{0}^{t}∫_{0}^{1} (\frac{λu_{y}^{2}}{vθ^{1/2}} + \frac{κθ_{y}^{2}}{2vθ^{3/2}} ) dy ds ≤ M.

Then

\begin{aligned}∫_{0}^{t} max_{y} θ^{(2q+3)/2}ds & ≤ M + M ∫_{0}^{t}∫_{0}^{1} θ^{(2q−3)/2}θ_{y}^{2} dy ds \\ & ≤ M + M ∫_{0}^{t}∫_{0}^{1} \frac{κθ_{y}^{2}}{θ^{3/2}} dy ds ≤ M. \end{aligned}

Step 4. Now we estimate Z. Differentiate the equation (2.1b) with respect to t, multiply it by u_{t}, and then integrate to obtain

\frac{1}{2} ∫_{0}^{1} u_{t}^{2}dy + ∫_{0}^{t}∫_{0}^{1} (\frac{λ}{v}u_{yt} + (\frac{λ}{v})_{v} u_{y}^{2} − p_{t} − b · b_{t}) u_{yt} dy ds ≤ M,

from integration by parts and the initial-boundary condition (2.2). Using Lemmas 3.1 and 3.2 and (3.13), one has the following estimates:

∫_{0}^{t}∫_{0}^{1} \frac{λ}{v}u_{yt}^{2} dy ds ≥ M_{7} ∫_{0}^{t}∫_{0}^{1} u_{yt}^{2} dy ds,

\begin{aligned}∫_{0}^{t}∫_{0}^{1} (\frac{λ}{v})_{v} u_{y}^{2}u_{yt} dy ds & ≤ \frac{M_{7}}{8} ∫_{0}^{t}∫_{0}^{1} u_{yt}^{2} dy ds + M ∫_{0}^{t}∫_{0}^{1} u_{y}^{4} dy ds \\ & ≤ \frac{M_{7}}{8} ∫_{0}^{t}∫_{0}^{1} u_{yt}^{2} dy ds + M, \end{aligned}

and ∫_{0}^{t}∫_{0}^{1} p_{t}u_{yt} dy ds ≤ ∫_{0}^{t}∫_{0}^{1} (|p_{v}||u_{y}| + |p_{θ}||θ_{t}|) |u_{yt}| dy ds

≤ \frac{M_{7}}{8} ∫_{0}^{t}∫_{0}^{1} u_{yt}^{2} dy ds + M (1 + Θ^{2+2r}) ∫_{0}^{t}∫_{0}^{1} u_{y}^{2} dy ds

+ M ∫_{0}^{t}∫_{0}^{1} (1 + θ^{2r}) θ_{t}^{2} dy ds

≤ \frac{M_{7}}{8} ∫_{0}^{t}∫_{0}^{1} u_{yt}^{2} dy ds + M(X + Y + 1)

≤ \frac{M_{7}}{8} ∫_{0}^{t}∫_{0}^{1} u_{yt}^{2} dy ds + M + MZ^{β_{4}}.

From the equation (3.9),

|b_{t}|^{2} ≤ M (|b|^{2}u_{y}^{2} + |w_{y}|^{2} + |b_{yy}|^{2} + |b_{y}|^{2}v_{y}^{2}) ,

and then, by Lemmas 3.1–3.2,

∫_{0}^{t}∫_{0}^{1} |b_{t}|^{2} dy ds ≤ M ∫_{0}^{t}∫_{0}^{1} (|b|^{2}u_{y}^{2} + |w_{y}|^{2} + |b_{yy}|^{2} + |b_{y}|^{2}v_{y}^{2}) dy ds

≤ M max_{t} ∫_{0}^{1} |b_{y}|^{2}dy ∫_{0}^{t}∫_{0}^{1} u_{y}^{2} dy ds + M ∫_{0}^{t}∫_{0}^{1} (|w_{y}|^{2} + |b_{yy}|^{2}) dy ds

+ M ∫_{0}^{t} max_{y} |b_{y}|^{2} ∫_{0}^{1} v_{y}^{2} dy ds

≤ M + M ∫_{0}^{t}∫_{0}^{1} (|b_{y}|^{2} + |b_{yy}|^{2}) dy ds ≤ M.

(3.19)

Thus, we have

\begin{aligned}∫_{0}^{t}∫_{0}^{1} u_{yt}b · b_{t} dy ds & ≤ \frac{M_{7}}{8} ∫_{0}^{t}∫_{0}^{1} u_{yt}^{2} dy ds + M ∫_{0}^{t}∫_{0}^{1} |b_{t}|^{2} dy ds \\ & ≤ \frac{M_{7}}{8} ∫_{0}^{t}∫_{0}^{1} u_{yt}^{2} dy ds + M, \end{aligned}

and then

∫_{0}^{1} u_{t}^{2}dy + ∫_{0}^{t}∫_{0}^{1} u_{yt}^{2} dy ds ≤ M + MZ^{β_{4}}.

From the equation (2.1b),

u_{yy} = \frac{v}{λ} (u_{t} − p_{y} − b · b_{y} − (\frac{λ}{v})_{v} v_{y}u_{y}) ,

and then, by Lemma 3.2 and (3.12),

∫_{0}^{1} u_{yy}^{2}dy ≤ ∫_{0}^{1} (u_{t}^{2} + p_{v}^{2}v_{y}^{2} + p_{θ}^{2}θ_{y}^{2} + |b_{y}|^{2} + v_{y}^{2}u_{y}^{2}) dy

≤ M + MZ^{β_{4}} + MΘ^{2+2r} ∫_{0}^{1} v_{y}^{2}dy + MY + M max_{y,t} u_{y}^{2} ∫_{0}^{1} v_{y}^{2}dy

≤ M + MZ^{3/4} + MZ^{β_{4}},

that is, Z ≤ M + MZ^{3/4} + MZ^{β_{4}}. Therefore Z ≤ M since the exponents of Z on the right side of the above inequality are both less than one.

Then (3.18) implies X + Y ≤ M and the lemma follows from (3.11).

We now prove the positive lower bound of the temperature with the aid of a maximum principle.

Lemma 3.4.

\begin{matrix}|(u, w, b, u_{y}, w_{y}, b_{y}, v_{y})| + ∫_{0}^{1} (|w_{yy}|^{2} + |b_{yy}|^{2} + θ_{yy}^{2}) ≤ M, \\ M^{−1} ≤ θ(y, t) ≤ M. \end{matrix} (3.20)

Proof. From Lemmas 3.1–3.3,

\begin{matrix}u^{2} + |w|^{2} + |b|^{2} \\ ≤ M ∫_{0}^{1} (u^{2} + |w|^{2} + |b|^{2}) dy + M ∫_{0}^{1} (u_{y}^{2} + |w_{y}|^{2} + |b_{y}|^{2}) dy ≤ M, \end{matrix}

and |u_{y}| ≤ M follows from (3.12). Differentiate (2.1c) with respect to t and then multiply by w_{t} and integrate it to obtain

\frac{1}{2} ∫_{0}^{1} |w_{t}|^{2}dy + ∫_{0}^{t}∫_{0}^{1} \frac{μ}{v}|w_{yt}|^{2} dy ds

≤ M − ∫_{0}^{t}∫_{0}^{1} (\frac{μ}{v})_{v} v_{t}w_{y} · w_{yt} dy ds − ∫_{0}^{t}∫_{0}^{1} b_{t} · w_{yt} dy ds

≤ \frac{1}{2} ∫_{0}^{t}∫_{0}^{1} \frac{μ}{v}|w_{yt}|^{2}dyds + M ∫_{0}^{t}∫_{0}^{1} (u_{y}^{4} + |w_{y}|^{4} + |b_{t}|^{2}) dy ds

≤ \frac{1}{2} ∫_{0}^{t}∫_{0}^{1} \frac{μ}{v}|w_{yt}|^{2}dyds + M,

from integration by parts, the initial-boundary condition (2.2), Lemma 3.2, and (3.19). Then

∫_{0}^{1} |w_{t}|^{2}dy + ∫_{0}^{t}∫_{0}^{1} \frac{μ}{v}|w_{yt}|^{2} dy ds ≤ M.

From the equation (2.1c),

w_{yy} = \frac{v}{μ} (w_{t} − b_{y} − (\frac{μ}{v})_{v} v_{y}w_{y}) ,

and then

∫_{0}^{1} |w_{yy}|^{2}dy ≤ M ∫_{0}^{1} (|w_{t}|^{2} + |b_{y}|^{2} + v_{y}^{2}|w_{y}|^{2}) dy

≤ M + M max_{y} |w_{y}|^{2} ∫_{0}^{1} v_{y}^{2}dy

≤ M + M ∫_{0}^{1} |w_{y}|^{2}dy + \frac{1}{2} ∫_{0}^{1} |w_{yy}|^{2}dy,

which implies that

∫_{0}^{1} |w_{yy}|^{2}dy ≤ M + M ∫_{0}^{1} |w_{y}|^{2}dy ≤ M,

and

|w_{y}|^{2} ≤ M ∫_{0}^{1} |w_{y}|^{2}dy + M ∫_{0}^{1} |w_{yy}^{2}|dy ≤ M.

Rewrite the equation (3.14) into:

e_{θ}θ_{t} + θp_{θ}u_{y} − \frac{λ}{v}u_{y}^{2} − \frac{μ}{v}|w_{y}|^{2} − \frac{ν}{v} |b_{y}|^{2} = (\frac{κ}{v})_{v} v_{y}θ_{y} + \frac{κ_{θ}}{v} θ_{y}^{2} + \frac{κ}{v}θ_{yy}.

Then, by Lemmas 3.2 and 3.3, and an interpolation inequality on θ_{y}^{2},

∫_{0}^{t}∫_{0}^{1} \frac{κ^{2}}{v^{2}} θ_{yy}^{2} dy ds

≤ M ∫_{0}^{t}∫_{0}^{1} (θ_{t}^{2} + u_{y}^{2} + u_{y}^{4} + |w_{y}|^{4} + |b_{y}|^{4} + v_{y}^{2}θ_{y}^{2} + θ_{y}^{4}) dy ds

≤ M + M ∫_{0}^{t} max_{y} θ_{y}^{2} ∫_{0}^{1} (v_{y}^{2} + θ_{y}^{2}) dy ds

≤ M + M ∫_{0}^{t}∫_{0}^{1} θ_{y}^{2} dy ds + \frac{1}{2} ∫_{0}^{t}∫_{0}^{1} \frac{κ^{2}}{v^{2}} θ_{yy}^{2} dy ds,

which implies

∫_{0}^{t}∫_{0}^{1} \frac{κ^{2}}{v^{2}} θ_{yy}^{2} dy ds ≤ M + M ∫_{0}^{t}∫_{0}^{1} θ_{y}^{2} dy ds ≤ M.  (3.21)

From (3.4),

V (v(y, t))_{y} = V (v(y, 0))_{y} + u(y, t) − u_{0}(y) + ∫_{0}^{t}(p_{y} + b · b_{y})dy.

Then, using the interpolation inequalities on |b_{y}|^{2} and θ_{y}^{2}, Lemma 3.1–3.3, and (3.21), we have

v_{y}^{2} ≤ M(V (v)_{y})^{2} ≤ M + M ∫_{0}^{t} (|b_{y}|^{2} + p_{v}^{2}v_{y}^{2} + p_{θ}^{2}θ_{y}^{2}) ds
≤ M + M ∫_{0}^{t}∫_{0}^{1} (|b_{y}|^{2} + |b_{yy}|^{2} + θ_{y}^{2} + θ_{yy}^{2}) dy ds + M ∫_{0}^{t} v_{y}^{2}ds
≤ M + M ∫_{0}^{t} v_{y}^{2}ds,

which yields, from Gronwall’s inequality, v_{y}^{2} ≤ M. Differentiate the equation (2.1d) with respect to t and then multiply by (vb)_{t} and integrate to obtain

\frac{1}{2} ∫_{0}^{1}(vb)_{t}^{2}dy

≤ M − \frac{1}{2} ∫_{0}^{t}∫_{0}^{1} νb_{yt}^{2} dy ds

+ M ∫_{0}^{t}∫_{0}^{1} (|w_{t}|^{2} + u_{y}^{2}|b_{y}|^{2} + u_{yy}^{2} + v_{y}^{2}|b_{t}|^{2}) dy ds

≤ M − \frac{1}{2} ∫_{0}^{t}∫_{0}^{1} νb_{yt}^{2} dy ds + M ∫_{0}^{t} max_{y} |b_{y}|^{2} ∫_{0}^{1} u_{y}^{2} dy ds

+ M ∫_{0}^{t}∫_{0}^{1} |b_{t}|^{2} dy ds

≤ M − \frac{1}{2} ∫_{0}^{t}∫_{0}^{1} νb_{yt}^{2} dy ds + M ∫_{0}^{t}∫_{0}^{1} (|b_{y}|^{2} + |b_{yy}|^{2}) dy ds

≤ M − \frac{1}{2} ∫_{0}^{t}∫_{0}^{1} νb_{yt}^{2} dy ds,

which yields

∫_{0}^{1}(vb)_{t}^{2}dy + ∫_{0}^{t}∫_{0}^{1} νb_{yt}^{2} dy ds ≤ M.

From the equation (2.1d),

\begin{aligned}∫_{0}^{1} |b_{yy}|^{2}dy & ≤ M ∫_{0}^{1} ((vb)_{t}^{2} + |w_{y}|^{2} + v_{y}^{2}|b_{y}|^{2}) dy \\ & ≤ M + M ∫_{0}^{1} |b_{y}|^{2}dy ≤ M, \end{aligned}

and then

|b_{y}|^{2} ≤ M ∫_{0}^{1} |b_{y}|^{2}dy + M ∫_{0}^{1} |b_{yy}|^{2}dy ≤ M.

As a consequence, we have M^{−1} ≤ θ ≤ M from the maximum principle (see [27]) applied to (3.14) and the boundedness of (v_{y}, u_{y}, w_{y}, b_{y}), 0 < θ ≤ M (from (3.13)), and the positive lower bound of θ_{0}.

If the initial data is Hölder continuous, following the standard method and argument in [14, 19, 22] and the Schauder estimates (cf. [20, 11]), we obtain that, for any T ∈ (0, ∞),

∥v∥_{C^{1+α,1+α/2}([0,1]×[0,T])} ≤ M, ∥(u, w, b, θ)∥_{C^{2+α,1+α/2}([0,1]×[0,T])} ≤ M,

from the above apriori estimates. To achieve these estimates, we differentiate the equations in (2.1) and make the energy estimates. Using the embedding theorem and the estimates of parabolic equations in the class of Hölder functions, we obtain the desired Hölder continuity of the solutions indicated in the theorem (see [14, 19, 22] for the details). This completes the proof of Theorem 2.1.

4. WEAK SOLUTIONS WITH LARGE DISCONTINUOUS INITIAL DATA

Before we prove Theorem 2.2, we first introduce a well-known lemma (see [28, 29]).

Lemma 4.1. Let X_{0}, X, and X_{1} be Banach spaces such that

X_{0} ⊂ X ⊂ X_{1},

where the injections are continuous and (1) the injection X_{0} → X is compact;

(2) X_{j}, j = 0, 1, are reflexive.

Let T > 0 and α_{0}, α_{1} > 1 are constants. Consider the space

Y ≡ \{v ∈ L^{α_{0}}(0, T; X_{0}), \frac{dv}{dt} ∈ L^{α_{1}}(0, T; X_{1})\}

under the norm

∥v∥_{Y} = ∥v∥_{L^{α_{0}} (0,T;X_{0})} + ∥\frac{dv}{dt} ∥_{L^{α_{1}} (0,T;X_{1})}.

Then the injection of Y into L^{α_{0}}(0, T; X) is compact.

Proof of Theorem 2.2. For the discontinuous initial data, we first approximate them by

(v_{0}^{ϵ}, u_{0}^{ϵ}, w_{0}^{ϵ}, b_{0}^{ϵ}, θ_{0}^{ϵ}) ∈ C^{∞}(0, 1),

via mollification such that

C_{0}^{−1} ≤ v_{0}^{ϵ}(y) ≤ C_{0}, θ_{0}^{ϵ}(y) ≥ M_{0ϵ}^{−1}, ∥(u_{0}^{ϵ}, w_{0}^{ϵ}, b_{0}^{ϵ})∥_{L^{4}} + ∥θ_{0}^{ϵ}∥_{L^{2}} ≤ C_{0},

where C_{0} > 0 is independent of ϵ. Theorem 2.1 implies that there exist global smooth solutions (v^{ϵ}, u^{ϵ}, w^{ϵ}, b^{ϵ}, θ^{ϵ})(y, t), and Lemmas 3.1–3.4 imply that there exists C > 0 independent of ϵ and M_{ϵ} > 0 such that

\begin{matrix}C^{−1} ≤ 1/v^{ϵ}(y, t) ≤ C, θ^{ϵ}(y, t) ≥ M_{ϵ}^{−1}, \\ ∥(u^{ϵ}, w^{ϵ}, b^{ϵ})∥_{L^{2}(0,T;L^{4}∩H^{1})} + ∥θ^{ϵ}∥_{L^{2}(0,T;L^{2}∩H^{1})} ≤ C, \end{matrix}  (4.1)

where C > 0 is independent of ϵ.

Take X_{0} = H_{0}^{1}(Ω), X = L^{4}(Ω), X_{1} = H^{−1}(Ω) for (u, w, b), and X_{0} = H_{0}^{1}(Ω), X = L^{2}(Ω), X_{1} = H^{−1}(Ω) for θ, and α_{0}, α_{1} = 2. Then Lemma 4.1 and (4.1) imply that there exists a subsequence (still denoted by) (v^{ϵ}, u^{ϵ}, w^{ϵ}, b^{ϵ}, θ^{ϵ}) such that

(u^{ϵ}, w^{ϵ}, b^{ϵ}) → (u, w, b) in L^{2}(0, T; L^{4}); θ^{ϵ} → θ ≥ 0 in L^{2}(0, T; L^{2});

1/v^{ϵ} ⇀^{∗} 1/v in L^{∞}; (u^{ϵ}, w^{ϵ}, b^{ϵ}, θ^{ϵ}) ⇀ (u, w, b, θ) in L^{2}(0, T; H^{1}).

Then it is easy to check that the limit function (ρ, u, w, b, θ) is a weak solution satisfying the equations in (2.1) in the sense of distributions such that C^{−1} ≤ v ≤ C, θ ≥ 0, (u, w, b) ∈ L^{2}(0, T; L^{4} ∩ H^{1}), θ ∈ L^{2}(0, T; L^{2} ∩ H^{1}). This completes the proof of Theorem 2.2.

5. APPENDIX: EQUATIONS OF MAGNETOHYDRODYNAMICS

Magnetohydrodynamics (MHD) concerns the motion of conducting fluids (cf. gases) in an electromagnetic field with a very broad range of applications. The dynamic motion of the fluids and the magnetic field interacts strongly on each other. We must consider both the hydrodynamic and electrodynamic effects. The equations of magnetohydrodynamic flows have the following form ([4, 21, 23]):

ρ_{t} + div(ρu) = 0, x ∈ R^{3}, t > 0,

(ρu)_{t} + div(ρu ⊗ u) + ∇p = (∇ × H) × H + div(λ′(divu)I)

+div(μ(∇u + (∇u)^{⊤})),

E_{t} + div(u(E + p)) = div((u × H) × H + νH × (∇ × H))

+div(u(λ′(divu)I + μ(∇u + (∇u)^{⊤}))) + div(κ∇θ),

H_{t} − ∇ × (u × H) = −∇ × (ν∇ × H), divH = 0,

(5.1)

where ρ is the density, u ∈ R^{3} the velocity, H ∈ R^{3} the magnetic field, and θ the temperature; the total energy is

E = ρ (e + \frac{1}{2}|u|^{2}) + \frac{1}{2}|H|^{2},

with e being the internal energy and \frac{1}{2}|H|^{2} the magnetic energy; the equations of state

p = p(ρ, θ), e = e(ρ, θ)

relate the pressure p and the internal energy e with the density and temperature of the flow; I is the 3 × 3 identity matrix, and (∇u)^{⊤} is the transpose of the matrix ∇u; λ′ = λ′(ρ, θ) and μ = μ(ρ, θ) are the viscosity coefficients of the flow satisfying λ′ + 2μ > 0, ν = ν(ρ, θ) is the magnetic diffusivity (see [1]) acting as a magnetic diffusion coefficient of the magnetic field, κ = κ(ρ, θ) is the heat conductivity, and all these kinetic coefficients and the magnetic diffusivity are independent of the magnitude and direction of the magnetic field (see [23]). The magnetic permeability differs only slightly from unity (the difference is not important in magnetohydrodynamics) and therefore is taken to be 1, which does not appear in the equations. The fluid under consideration is a Newtonian fluid, i.e., the stress tensor

−pI + (μ(∇u + (∇u)^{⊤}) + λ′(divu)I)

is a linear function of the deformation tensor \frac{1}{2}(∇u+(∇u)^{⊤}). The viscosity and heat conduction terms describe the dissipative processes in magnetohydrodynamics.

Equations (5.1) of magnetohydrodynamics are based on neglecting the displacement current in Maxwell’s equation. As a consequence, the last equation in (5.1)

H_{t} − ∇ × (u × H) = −∇ × (ν∇ × H)  (5.2)

is called the induction equation. The Maxwell equations governing the electromagnetic fields are:

ε_{0}E_{t} − ∇ × H + J = 0,  (5.3)
H_{t} + ∇ × E = 0,  div H = 0,  (5.4)

where ε_{0} is the dielectric constant, and the current density J is given by Oham’s law:

J = \frac{1}{ν} (E + u × H).

In magnetohydrodynamics, the displacement current E_{t} can be neglected (see [21, 23]), that is, we can set ε_{0}E_{t} = 0 in (5.3), and then J = ∇ × H. Therefore, the electric field can be written in terms of the magnetic field H and the velocity u,

E = ν∇ × H − u × H.  (5.5)

Then (5.2) is obtained by substituting (5.5) into (5.4). We see now, although the electric field E does not appear in (5.1), that it is indeed induced according to (5.5) by the moving conductive flow in the magnetic field.

Consider a three-dimensional MHD flow with spatial variables x = (x, x_{2}, x_{3}) which is moving in the x direction and uniform in the transverse direction (x_{2}, x_{3}):

\begin{matrix}ρ = ρ(x, t), θ = θ(x, t); u = (u, w)(x, t), w = (u_{2}, u_{3}); \\ H = (b_{1}, b)(x, t), b = (b_{2}, b_{3}), \end{matrix}  (5.6)

where u and b_{1} are the longitudinal velocity and longitudinal magnetic field, respectively; w and b are the transverse velocity and transverse magnetic field, respectively. The magnetohydrodynamic flow satisfies (5.1). With this special structure (5.6), equations (5.1) are reduced to (1.1) with the constant longitudinal magnetic field b_{1} and λ = λ′ + 2μ > 0. We take b_{1} = 1. The term \frac{1}{2}|b|^{2} appears in the second equation of (1.1) because of this equality: (∇ × H) × H = (−\frac{1}{2}|b|^{2}, b)_{x}^{⊤}. System (1.1) can be verified by direct calculations.

ACKNOWLEDGMENTS

Gui-Qiang Chen’s research was supported in part by the National Science Foundation through grants DMS-9971793, INT-9987378, and INT-9726215. Dehua Wang’s research was supported in part by the National Science Foundation and the Office of Naval Research.

REFERENCES

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