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Regularity of large solutions for the compressible magnetohydrodynamic equations

Xin Liu1*, Yuming Qin2 and Xiaozhen Peng3

Author Affiliations

1 College of Information Sciences and Technology, Donghua University, Songjiang Shanghai, 201620, People's Republic of China

2 Department of Applied Mathematics, College of Sciences, Donghua University, Songjiang Shanghai 201620, People's Republic of China

3 Songyin School, Jinshan Shanghai, 201504, People's Republic of China

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Boundary Value Problems 2011, 2011:30  doi:10.1186/1687-2770-2011-30

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/30


Received:10 June 2011
Accepted:10 October 2011
Published:10 October 2011

© 2011 Liu et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

Keywords:
magnetohydrodynamics (MHD); global solutions; regularity; initial-boundary value problem

1 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, 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:

v t - u y = 0 , (1.1)

u t + ( p + 1 2 b 2 ) y = λ u y v y , (1.2)

w t - b y = μ w y v y , (1.3)

( v b ) t - w y = ν b y v y , (1.4)

E t + u ( p + 1 2 b 2 ) - w b y = λ u u y + μ w w y + ν b b y + κ θ y v y . (1.5)

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

( v , u , w , b , θ ) | t = 0 = ( v 0 , u 0 , w 0 , b 0 , θ ) ( y ) , y Ω , (1.6)

( u , w , b , θ y ) | Ω = 0 . (1.7)

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

p = R θ v , e = c v θ ,

where R is the gas constant and cv 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 + θr) ≤ 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

E = e + 1 2 ( u 2 + w 2 ) + 1 2 v b 2 .

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

e = c v θ , σ = R θ v + μ u y v , Q = - κ θ y v , (1.8)

with suitable positive constants cv, 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)

σ ( v , θ , u y ) = - p ( v , θ ) + μ ( v , θ ) v u y (1.9)

satisfying Fourier's law of heat flux

Q ( v , θ , θ y ) = - κ ( v , θ ) v θ y (1.10)

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

k 1 ( 1 + θ q ) κ ( ρ , θ ) k 2 ( 1 + θ q ) , k 1 ( 1 + θ q ) | κ ρ ( ρ , θ ) | k 2 ( 1 + θ q ) ,

for some q > 5 2 , 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

C - 1 ( 1 + θ q ) κ ( ρ , θ ) C ( 1 + θ q )

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

κ ( ρ , θ ) κ ( ρ ) > C ρ ,

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 H + i ( i =  2 , 4 ) 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 H + i 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:

H + 1 = ( v , u , w , b , θ ) ( H 1 ( Ω ) ) 7 : v ( x ) > 0 , θ ( x ) > 0 , x Ω , u ( 0 ) = u ( 1 ) = 0 , w ( 0 ) = w ( 1 ) = b ( 0 ) = b ( 1 ) = 0 , H + i = ( v , u , w , b , θ ) ( H i ( Ω ) ) 7 : v ( x ) > 0 , θ ( x ) > 0 , x Ω , u ( 0 ) = u ( 1 ) = 0 , w ( 0 ) = w ( 1 ) = b ( 0 ) = b ( 1 ) = 0 , θ ( 0 ) = θ ( 1 ) = 0 , i = 2 , 4 .

The notation in this paper will be stated as follows:

Lp, 1 ≤ p ≤ +∞, Wm, p, m N, H1 = W1,2, H 0 1 = W 0 1 , 2 denote the usual (Sobolev) spaces on Ω. In addition, ||·||B denotes the norm in the space B, we also put = L 2 ( Ω ) . Constants Ci(i = 1, 2, 3, 4) depend on the H + i 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 ( v 0 , u 0 , w 0 , b 0 , θ 0 ) H + 2 and e, p, and κ are C3 functions. Then, the problem (1.1) -(1.7) admits a unique global solution ( v ( t ) , u ( t ) , w ( t ) , b ( t ) , θ ( t ) ) H + 2 such that for any T > 0,

v ( t ) v ¯ H 2 2 + u ( t ) H 2 2 + w ( t ) H 2 2 + b ( t ) H 2 2 + θ ( t ) θ ¯ H 2 2 + u t ( t ) 2 + w t ( t ) 2 + b t ( t ) 2 + θ t ( t ) 2 + 0 t ( v v ¯ H 2 2 + u H 3 2 + w H 3 2 + b H 3 2 + θ θ ¯ H 3 2 + u t y 2 + w t y 2 + b t y 2 + θ t y 2 ) ( s ) d s C 2 , t [ 0 , T ] , (1.11)

where v ̄ = 0 1 v d y = 0 1 v 0 d y , constant θ ̄ > 0 is determined by

e ( v ̄ , θ ̄ ) = 0 1 1 2 ( u 0 2 + | w 0 | 2 + v 0 | b 0 | 2 ) + e ( v 0 , θ 0 ) ( y ) d y .

Theorem 1.2 Assume that the initial data ( v 0 , u 0 , w 0 , b 0 , θ 0 ) H + 4 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 ( v ( t ) , u ( t ) , w ( t ) , b ( t ) , θ ( t ) ) H + 4 such that for any T > 0,

v ( t ) v ¯ H 4 2 + u ( t ) H 4 2 + w ( t ) H 4 2 + b ( t ) H 4 2 + θ ( t ) θ ¯ H 4 2 + u t t ( t ) 2 + w t t ( t ) 2 + b t t ( t ) 2 + u t ( t ) H 2 2 + w t ( t ) H 2 2 + b t ( t ) H 2 2 + θ t ( t ) H 2 2 + θ t t ( t ) 2 + 0 t ( v v ¯ H 4 2 + u H 5 2 + w H 5 2 + b H 5 2 + θ θ ¯ H 5 2 + u t H 3 2 + w t H 3 2 + b t H 3 2 + θ t H 3 2 + u t t H 1 2 + w t t H 1 2 + b t t H 1 2 + θ t t H 1 2 ) ( s ) d s C 4 , t [ 0 , T ] . (1.12)

2 Proof of Theorem 1.1

In this section, we study the global existence of problem (1.1)-(1.7) in H + 2 by establishing a series of priori estimates. Without loss of generality, we take cv = R = 1. We begin with the following lemma.

Lemma 2.1 Assume that the initial data ( v 0 , u 0 , w 0 , b 0 , θ 0 ) H + 1 and e, p, and κ are C2 functions on 0 < v < +∞ and 0 ≤ θ < +∞ and there exists a positive constant C0 such that

0 < C 0 - 1 v 0 ( y ) C 0 , 0 < C 0 - 1 θ 0 ( y ) C 0 .

Then, for the initial data ( v 0 , u 0 , w 0 , b 0 , θ 0 ) H + 1 , the problem (1.1) -(1.7) admits a unique global solution ( v ( t ) , u ( t ) , w ( t ) , b ( t ) , θ ( t ) ) H + 1 such that for any T > 0

0 < C 1 - 1 v ( y , t ) C 1 , 0 < C 1 - 1 θ ( y , t ) C 1 , ( y , t ) [ 0 , 1 ] × [ 0 , T ] (2.1)

and for any t ∈ [0, T],

v ( t ) v ¯ H 1 2 + u ( t ) H 1 2 + w ( t ) H 1 2 + b ( t ) H 1 2 + θ ( t ) θ ¯ H 1 2 + 0 t ( v v ¯ H 1 2 + u H 2 2 + w H 2 2 + b H 2 2 + θ θ ¯ H 2 2 + u t 2 + w t 2 + b t 2 + θ t 2 ) ( s ) d s C 1 . (2.2)

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

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

u t ( t ) 2 + w t ( t ) 2 + b t ( t ) 2 + θ t ( t ) 2 + 0 t ( u t y 2 + w t y 2 + b t y 2 + θ t y 2 ) ( s ) d s C 2 , t [ 0 , T ] . (2.3)

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

u t ( t ) 2 + 0 t u t y ( s ) 2 d s ε 0 t u t y ( s ) 2 d s + C 2 0 t ( θ t ( s ) 2 + b b t ( s ) 2 + u y ( s ) L 4 4 ) d s ε 0 t u t y ( s ) 2 d s + C 2 0 t ( u y ( s ) L 2 + θ t ( s ) 2 + b t ( s ) 2 ) d s C 2 + ε 0 t u t y ( s ) 2 d s ,

which implies

u t ( t ) 2 + 0 t u t y ( s ) 2 d s C 2 . (2.4)

Analogously, we have

w t ( t ) 2 + b t ( t ) 2 + θ t ( t ) 2 + 0 t ( w t y 2 + b t y 2 + θ t y 2 ) ( s ) d s C 2 . (2.5)

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

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

u y y ( t ) 2 + w y y ( t ) 2 + b y y ( t ) 2 + θ y y ( t ) 2 + 0 t ( u y y y 2 + w y y y 2 + b y y y 2 + θ y y y 2 ) ( s ) d s C 2 , t [ 0 , T ] . (2.6)

Proof. Equation (1.2) can be rewritten as

u t = - λ θ y v + λ θ v y v - λ u y v y v 2 + λ u y y v - b b y . (2.7)

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

u y y ( t ) C 2 ( | | u t ( t ) | | + | | θ y ( t ) | | + | | b b y ( t ) | | + | | θ v y ( t ) | | + | | v y u y ( t ) | | ) C 2 ( | | u t ( t ) | | + | | θ y ( t ) | | + θ ( t ) L | | v y ( t ) | | + | | b y ( t ) | | 2 + | | u y ( t ) | | L | | v y ( t ) | | ) ε | | u y y ( t ) | | + C 2 ( | | u t ( t ) | | + 1 ) ,

which leads to

u y y ( t ) C 2 , 0 t u y y y ( s ) 2 d s C 2 0 t u t y ( s ) 2 d s C 2 . (2.8)

Similarly, we derive

w y y ( t ) + b y y ( t ) + θ y y ( t ) C 2 ( w t ( t ) + b t ( t ) + θ t ( t ) + 1 ) C 2 , (2.9)

0 t ( w y y y 2 + b y y y 2 + θ y y y 2 ) ( s ) d s C 2 . (2.10)

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

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

v y y ( t ) 2 + 0 t v y y ( s ) 2 d s C 2 , t [ 0 , T ] . (2.11)

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

λ d d t v y y v + θ v 2 v y y = u t y + E ( y , t ) , (2.12)

where

E ( y , t ) = θ y y v + 2 v y ( λ u y y - θ y ) v 2 + 2 v y 2 ( θ - λ u y ) v 3 + b b y y + b y 2 .

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

v y y v ( t ) 2 + C 1 1 0 t v y y v ( s ) 2 d s 1 4 C 1 0 t v y y v ( s ) 2 d s + C 2 0 t ( u t y 2 + θ y y 2 + v y u y y 2 + v y L 4 4 + u y v y 2 2 + b y L 4 4 + b L 2 b y y 2 ) ( s ) d s 1 2 C 1 0 t v y y v ( s ) 2 d s + C 2 0 t ( v y 2 + u t y 2 + u y y L 2 + θ y y 2 + b y y 2 ) ( s ) d s ,

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.

3 Proof of Theorem 1.2

In this section, we study the global existence of problem (1.1)-(1.7) in H + 4 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:

u t y ( y , 0 ) + w t y ( y , 0 ) + w t y ( y , 0 ) + θ t y ( y , 0 ) C 3 , (3.1)

u t t ( y , 0 ) + w t t ( y , 0 ) + b t t ( y , 0 ) + θ t t ( y , 0 ) + u t y y ( y , 0 ) + w t y y ( y , 0 ) + b t y y ( y , 0 ) + θ t y y ( y , 0 ) C 3 . (3.2)

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

u t ( t ) C 3 ( v y ( t ) + θ y ( t ) + u y y ( t ) + u y ( t ) L v y ( t ) + b ( t ) L b y ( t ) ) C 3 ( v y ( t ) + θ y ( t ) + u y y ( t ) + b y ( t ) ) .

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

u t y ( t ) C 3 ( v y ( t ) H 1 + θ y ( t ) H 1 + u y ( t ) H 2 + b y ( t ) H 1 ) , (3.3)

or

u y y y ( t ) C 3 ( v y ( t ) H 1 + θ y ( t ) H 1 + b y ( t ) H 1 + u t y ( t ) ) . (3.4)

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

u t y y ( t ) C 3 ( v y ( t ) H 2 + θ y ( t ) H 2 + u y ( t ) H 3 + b y ( t ) H 2 ) , (3.5)

or

u y y y y ( t ) C 3 ( v y ( t ) H 2 + θ y ( t ) H 2 + b y ( t ) H 2 + u t y y ( t ) ) . (3.6)

Similarly, we have

w t ( t ) C 3 ( w y ( t ) H 1 + b y ( t ) + v y ( t ) ) , w t y ( t ) C 3 ( w y ( t ) H 2 + b y ( t ) H 1 + v y ( t ) H 1 ) , (3.7)

or

w y y y ( t ) C 3 ( b y ( t ) H 1 + v y ( t ) H 1 + b t y ( t ) ) , (3.8)

w t y y ( t ) C 3 ( w y ( t ) H 3 + b y ( t ) H 2 + v y ( t ) H 2 ) , (3.9)

or

w y y y y ( t ) C 3 ( b y ( t ) H 2 + v y ( t ) H 2 + w t y y ( t ) ) , (3.10)

b t C 3 ( b y H 1 + w y + v y ) , b t y ( t ) C 3 ( b y ( t ) H 2 + w y ( t ) H 1 + v y ( t ) H 1 ) , (3.11)

or

b y y y ( t ) C 3 ( w y ( t ) H 1 + v y ( t ) H 1 + b t y ( t ) ) , (3.12)

b t y y ( t ) C 3 ( b y ( t ) H 3 + w y ( t ) H 2 + v y ( t ) H 2 ) , (3.13)

or

b y y y y ( t ) C 3 ( w y ( t ) H 2 + v y ( t ) H 2 + b t y y ( t ) ) , (3.14)

θ t ( t ) C 3 ( u y ( t ) + v y ( t ) + θ y y ( t ) + u y ( t ) L u y ( t ) + w y ( t ) L w y ( t ) + b y ( t ) L b y ( t ) + θ y ( t ) L θ y ( t ) ) , C 3 ( θ y y ( t ) + u y y ( t ) + w y y ( t ) + b y y ( t ) ) , (3.15)

θ t y ( t ) C 3 ( θ t ( t ) + θ y ( t ) H 2 + v y ( t ) H 1 + u y ( t ) H 1 + w y ( t ) H 1 + b y ( t ) H 1 ) , (3.16)

or

θ y y y ( t ) C 3 ( v y ( t ) H 1 + u y ( t ) H 1 + w y ( t ) H 1 + b y ( t ) H 1 + θ t y ( t ) ) , (3.17)

θ t y y ( t ) C 3 ( θ y ( t ) H 3 + v y ( t ) H 2 + u y ( t ) H 2 + w y ( t ) H 2 + b y ( t ) H 2 ) , (3.18)

or

θ y y y y ( t ) C 3 ( v y ( t ) H 2 + u y ( t ) H 2 + w y ( t ) H 2 + b y ( t ) H 2 + θ t y y ( t ) ) . (3.19)

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

u t t ( t ) C 3 ( v y ( t ) H 2 + u y ( t ) H 3 + b y ( t ) H 2 + θ y ( t ) H 2 ) . (3.20)

Similarly, we can conclude

w t t ( t ) C 3 ( v y ( t ) H 2 + b y ( t ) H 2 + w y ( t ) H 3 ) , (3.21)

b t t ( t ) C 3 ( v y ( t ) H 2 + b y ( t ) H 3 + w y ( t ) H 2 ) , (3.22)

θ t t ( t ) C 3 ( v y ( t ) H 2 + u y ( t ) H 2 + b y ( t ) H 2 + w y ( t ) H 2 + θ y ( t ) H 3 ) . (3.23)

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],

u t t ( t ) 2 + 0 t u t t y ( s ) 2 d s C 3 + C 3 0 t ( b t y y 2 + θ t y y 2 ) ( s ) d s , (3.24)

w t t ( t ) 2 + b t t ( t ) 2 + 0 t ( w t t y ( s ) 2 + b t t y ( s ) 2 ) d s C 3 + C 3 0 t ( b t y y ( s ) 2 + w t y y ( s ) 2 ) d s , (3.25)

θ t t ( t ) 2 + 0 t θ t t y ( s ) 2 d s C 3 + C 2 ε - 1 0 t θ t y y ( s ) 2 d s + C 1 ε 0 t ( u t y y 2 + u t t y 2 + w t y y 2 + w t t y 2 + b t y y 2 + b t t y 2 ) ( s ) d s . (3.26)

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

1 2 d d t 0 1 u t t 2 ( y , t ) d y λ u t t y ( t ) 2 + C 4 ( θ t t ( t ) + u y y ( t ) + θ t u y ( t ) + b b t t ( t ) + b t 2 ( t ) 2 + u t y ( t ) ) u t t y ( t ) C 1 1 u t t y ( t ) 2 + C 4 ( θ t t ( t ) 2 + b t t ( t ) 2 + u t y ( t ) 2 + u y ( t ) 2 + θ t ( t ) 2 ) (3.27)

Thus, using Theorem 1.1 and Lemma 3.1, we get

u t t ( t ) 2 + 0 t u t t y ( s ) 2 d s C 3 + C 3 0 t ( b t y y 2 + θ t y y 2 ) ( s ) d s .

Analogously, we obtain

w t t ( t ) 2 + b t t ( t ) 2 + 0 t ( w t t y 2 + b t t y 2 ) ( s ) d s C 3 + C 3 0 t ( w t y y 2 + b t y y 2 ) ( s ) d s .

Equation (1.5) can be rewritten as

( c v θ ) t + p u y = κ θ y v y + λ u y 2 + μ | w y | 2 + ν | b y | 2 v . (3.28)

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

1 2 d d t 0 1 c v θ t t ( y , t ) d y = - 0 1 κ θ y v t t θ t t y ( y , t ) d y - 0 1 p - λ u y v u t t y θ t t ( y , t ) d y + 0 1 μ w y v w t t y + ν b y v b t t y θ t t ( y , t ) d y - 2 0 1 p t - ( λ u y v ) t u t y θ t t ( y , t ) d y + 2 0 1 ( μ w y v ) t w t y + ( ν b y v ) t b t y θ t t ( y , t ) d y + 0 1 - p + λ u y v t t u y θ t t ( y , t ) d y + 0 1 ( μ w y v ) t t w y + ( ν b y v ) t t b y θ t t ( y , t ) d y = B 1 + B 2 + B 3 + B 4 + B 5 + B 6 + B 7 .

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

B 1 C 1 θ t t y ( t ) 2 + C 2 θ t y ( t ) L ( u y ( t ) 2 + θ t ( t ) 2 ) θ t t y ( t ) + C 2 ( κ v ) t t ( t ) 2 θ y ( t ) L θ t t y ( t ) 2 C 1 θ t t y ( t ) 2 + C 2 ( u y ( t ) 2 + θ t ( t ) 2 + u t y ( t ) 2 + θ t y ( t ) 2 + θ t t ( t ) 2 + θ t y y ( t ) 2 ) , B 2 ε u t t y ( t ) 2 + C 2 ε 1 θ t t ( t ) 2 , B 3 ε ( w t t y ( t ) 2 + b t t y ( t ) 2 ) + C 2 ε 1 θ t t ( t ) 2

and

B 4 C 1 0 1 ( | θ t | + | u y | + | u t y | + | u y | 2 ) | θ t t u t y | ( y , t ) d y C 2 u t y ( t ) | | 1 2 u t y y ( t ) | | 1 2 ( θ t ( t ) + u y ( t ) + u t y ( t ) ) θ t t ( t )

which implies

0 t B 4 d s C 2 sup 0 s t θ t t ( s ) 0 t u t y ( s ) | | 2 d s 1 4 0 t u t y y ( s ) | | 2 d s 1 4 × 0 t ( u t y | | 2 + θ t | | 2 + u y | | 2 ) ( s ) d s 1 2 ε sup 0 s t θ t t ( s ) | | 2 + 0 t u t y y ( s ) | | 2 d s + C 3 ε - 3 .

B 5 C 1 0 1 ( | w y | 2 + | w t y | ) | w t y | + ( | w y | 2 + | w t y | ) | w t y | | θ t t | ( y , t ) d y C 2 w t y ( t ) | | L ( w y ( t ) | | 2 + w t y ( t ) ) θ t t ( t ) + C 2 w t y ( t ) | | L ( b y ( t ) | | 2 + b t y ( t ) ) θ t t ( t )

which implies

0 t B 5 d s ε ( sup 0 s t θ t t ( s ) 2 + 0 t ( w t y y 2 + w t y y 2 ( s ) d s ) + C 3 ε 3 , B 6 C 2 u y ( t ) | | L θ t t ( t ) [ ( θ t ( t ) | | L + u y ( t ) | | L ) ( θ t ( t ) + u y ( t ) ) + θ t t ( t ) + u t y ( t ) + u y ( t ) + u t t y ( t ) ] C 2 θ t t ( t ) ( θ t ( t ) + u y ( t ) | | H 1 + θ t y ( t ) + θ t t ( t ) + u t y ( t ) + u t t y ( t ) ) ε u t t y ( t ) 2 + C 2 ε 1 ( θ t ( t ) 2 + u y ( t ) | | H 1 2 + θ t y ( t ) 2 + θ t t ( t ) 2 + u t y ( t ) 2 ) , B 7 C 2 w y ( t ) | | L θ t t ( t ) ( w t t y ( t ) + w t y ( t ) + w y ( t ) ) + C 2 b y ( t ) | | L θ t t ( t ) × ( b t t y ( t ) + b t y ( t ) + b y ( t ) ) C 2 θ t t ( t ) ( w t y ( t ) + w y ( t ) | | H 1 + w t t y ( t ) + b t y ( t ) + b y ( t ) | | H 1 + b t t y ( t ) ) ε ( w t t y ( t ) 2 + b t t y ( t ) 2 ) + C 2 ε 1 ( θ t t ( t ) 2 + w y ( t ) | | H 1 2 + w t y ( t ) 2 + b y ( t ) | | H 1 2 + b t y ( t ) 2 ) .

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

θ t t ( t ) 2 + 0 t θ t t y ( s ) 2 d s C 2 ε 1 0 t ( θ t y y ( s ) 2 + θ t t ( s ) 2 ) d s + C 3 ε 3 + C 1 ε [ sup 0 s t θ t t ( s ) 2 + 0 t ( u t y y 2 + u t t y 2 + w t y y 2 + w t t y 2 + b t y y 2 + b t t y 2 ) ( s ) d s ] . (3.29)

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],

u t y ( t ) 2 + 0 t u t y y ( s ) 2 d s C 3 ε - 6 + C 2 ε 2 0 t ( b t y y 2 + θ t y y 2 + u t t y 2 ) ( s ) d s , (3.30)

w t y ( t ) 2 + b t y ( t ) 2 + 0 t ( w t y y 2 + b t y y 2 ) ( s ) d s C 3 ε - 6 + C 2 ε 2 0 t ( w t y y 2 + b t t y 2 ) ( s ) d s , (3.31)

θ t y ( t ) 2 + 0 t θ t y y ( s ) 2 d s C 3 ε - 6 + C 2 ε 2 0 t ( b t y y 2 + u t y y 2 + w t y y 2 + θ t t y 2 + θ y y y 2 θ t y 2 ) ( s ) d s . (3.32)

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

1 2 d d t u t y ( t ) 2 = D 0 ( y , t ) + D 1 ( t ) , (3.33)

where

D 0 ( y , t ) = σ t y u t y | y = 0 y = 1 , D 1 ( t ) = - 0 1 σ t y u t y y d y , σ = - p + 1 2 | b | 2 - λ u y v .

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

D 0 C 1 [ ( u y ( t ) L + θ t ( t ) L ) ( v y ( t ) L + θ y ( t ) L ) + b t ( t ) L b y ( t ) L + b t y ( t ) L b ( t ) L + θ t y ( t ) L + u y ( t ) L 2 + u y ( t ) L u y y ( t ) L + u t y ( t ) L v y ( t ) L + u t y y ( t ) L ] u t y ( t ) L C 3 ( D 01 + D 01 ) u t y ( t ) 1 2 u t y y ( t ) 1 2 , (3.34)

where

D 0 1 = u y ( t ) H 2 + θ t ( t ) + θ t y ( t ) + b t ( t ) + b t y ( t ) , D 0 2 = θ t y ( t ) 1 2 θ t y y ( t ) 1 2 + u t y y ( t ) 1 2 u t y y y ( t ) 1 2 + u t y y ( t ) + u t y ( t ) 1 2 u t y y ( t ) 1 2 + b t y ( t ) 1 2 b t y y ( t ) 1 2 .

Using the Young inequality several times, we derive

C 3 D 0 1 u t y ( t ) 1 2 u t y y ( t ) 1 2 ε 2 2 u t y y ( t ) 2 + C 3 ε - 2 3 ( u t y ( t ) 2 + u y ( t ) H 2 2 + θ t ( t ) 2 + θ t y ( t ) 2 + b t ( t ) 2 + b t y ( t ) 2 ) (3.35)

and

C 3 D 0 2 u t y ( t ) 1 2 u t y y ( t ) 1 2 ε 2 2 u t y y ( t ) 2 + ε 2 ( u t y y y ( t ) 2 + b t y y ( t ) 2 + θ t y y ( t ) 2 ) + C 3 ε - 6 ( u t y ( t ) 2 + θ t y ( t ) 2 + b t y ( t ) 2 ) . (3.36)

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

D 0 ε 2 ( u t y y y ( t ) 2 + u t y y ( t ) 2 + b t y y ( t ) 2 + θ t y y ( t ) 2 ) + C 3 ε - 6 ( u t y ( t ) 2 + θ t y ( t ) 2 + b t y ( t ) 2 + θ t ( t ) 2 + u y ( t ) H 2 2 + b t ( t ) 2 ) ,

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

0 t D 0 d s ε 2 0 t ( u t y y y 2 + u t y y 2 + b t y y 2 + θ t y y 2 ) ( s ) d s + C 3 ε - 6 . (3.37)

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

D 1 ( 2 C 3 ) - 1 u t y y ( t ) 2 + C 3 ( u t y ( t ) 2 + b t y ( t ) 2 + θ t ( t ) H 1 2 + u y ( t ) H 1 2 ) , (3.38)

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,

u t y ( t ) 2 + 0 t u t y y ( s ) 2 d s C 3 ε - 6 + C 2 ε 2 0 t ( b t y y 2 + θ t y y 2 + u t y y y 2 ) ( s ) d s . (3.39)

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

u t y y y ( t ) C 1 u t t y ( t ) + C 2 ( u y y ( t ) H 2 2 + θ y ( t ) H 1 2 + v y ( t ) H 1 2 + b y ( t ) H 1 2 + θ t ( t ) H 2 2 + b t ( t ) H 2 2 ) . (3.40)

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],

u t t ( t ) 2 + u t y ( t ) 2 + w t t ( t ) 2 + w t y ( t ) 2 + b t t ( t ) 2 + b t y ( t ) 2 + θ t t ( t ) 2 + θ t y ( t ) 2 + 0 t ( u t t y 2 + u t y y 2 + w t t y 2 + w t y y 2 + b t t y 2 + b t y y 2 + θ t t y 2 + θ t y y 2 ) ( s ) d s C 4 ,