SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Global behavior of 1D compressible isentropic Navier-Stokes equations with a non-autonomous external force

Lan Huang* and Ruxu Lian

Author Affiliations

College of Mathematics and Information Science, North China University of Water Sources and Electric Power, Zhengzhou 450011, People's Republic of PR China

For all author emails, please log on.

Boundary Value Problems 2011, 2011:43  doi:10.1186/1687-2770-2011-43

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/43


Received:14 June 2011
Accepted:3 November 2011
Published:3 November 2011

© 2011 Huang and Lian; 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 study a free boundary problem for compressible Navier-Stokes equations with density-dependent viscosity and a non-autonomous external force. The viscosity coefficient μ is proportional to ρθ with 0 < θ < 1, where ρ is the density. Under certain assumptions imposed on the initial data and external force f, we obtain the global existence and regularity. Some ideas and more delicate estimates are introduced to prove these results.

Keywords:
Compressible Navier-Stokes equations; Viscosity; Regularity; Vacuum

1 Introduction

We study a free boundary problem for compressible Navier-Stokes equations with density-dependent viscosity and a non-autonomous external force, which can be written in Eulerian coordinates as:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M1">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M2">View MathML</a>

(1.2)

with initial data

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M3">View MathML</a>

(1.3)

where ρ = ρ (ξ,τ), u = u(ξ,τ), P = P(ρ) and f = f(ξ,t) denote the density, velocity, pressure and a given external force, respectively, μ = μ(ρ) is the viscosity coefficient. a(τ) and b(τ) are the free boundaries with the following property:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M4">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M5">View MathML</a>

(1.5)

The investigation in [1] showed that the continuous dependence on the initial data of the solutions to the compressible Navier-Stokes equations with vacuum failed. The main reason for the failure at the vacuum is because of kinematic viscosity coefficient being independent of the density. On the other hand, we know that the Navier-Stokes equations can be derived from the Boltzmann equation through Chapman-Enskog expansion to the second order, and the viscosity coefficient is a function of temperature. For the hard sphere model, it is proportional to the square-root of the temperature. If we consider the isentropic gas flow, this dependence is reduced to the dependence on the density function by using the second law of thermal dynamics.

For simplicity of presentation, we consider only the polytropic gas, i.e. P(ρ) = γ with A > 0 being constants. Our main assumption is that the viscosity coefficient μ is assumed to be a functional of the density ρ, i.e. μ = θ, where c and θ are positive constants. Without loss of generality, we assume A = 1 and c = 1.

Since the boundaries x = a(τ) and x = b(τ) are unknown in Euler coordinates, we will convert them to fixed boundaries by using Lagrangian coordinates. We introduce the following coordinate transformation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M6">View MathML</a>

(1.6)

then the free boundaries ξ = a(τ) and ξ = b(τ) become

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M7">View MathML</a>

(1.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M8">View MathML</a> is the total initial mass, and without loss of generality, we can normalize it to 1. So in terms of Lagrangian coordinates, the free boundaries become fixed. Under the coordinate transformation, Eqs. (1.1)-(1.2) are now transformed into

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M9">View MathML</a>

(1.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M10">View MathML</a>

(1.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M11">View MathML</a>. The boundary conditions (1.4)-(1.5) become

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M12">View MathML</a>

(1.10)

and the initial data (1.3) become

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M13">View MathML</a>

(1.11)

Now let us first recall some previous works in this direction. When the external force f ≡ 0, there have been many works (see, e.g., [2-9]) on the existence and uniqueness of global weak solutions, based on the assumption that the gas connects to vacuum with jump discontinuities, and the density of the gas has compact support. Among them, Liu et al. [4] established the local well-posedness of weak solutions to the Navier-Stokes equations; Okada et al. [5] obtained the global existence of weak solutions when 0 < θ < 1/3 with the same property. This result was later generalized to the case when 0 < θ < 1/2 and 0 < θ < 1 by Yang et al. [7] and Jiang et al. [3], respectively. Later on, Qin et al. [8,9] proved the regularity of weak solutions and existence of classical solution. Fang and Zhang [2] proved the global existence of weak solutions to the compressible Navier-Stokes equations when the initial density is a piece-wise smooth function, having only a finite number of jump discontinuities.

For the related degenerated density function and viscosity coefficient at free boundaries, see Yang and Zhao [10], Yang and Zhu [11], Vong et al. [12], Fang and Zhang [13,14], Qin et al. [15], authors studied the global existence and uniqueness under some assumptions on initial data.

When f ≠ 0, Qin and Zhao [16] proved the global existence and asymptotic behavior for γ = 1 and μ = const with boundary conditions u(0,t) = u(1,t) = 0; Zhang and Fang [17] established the global behavior of the Equations (1.1)-(1.2) with boundary conditions u(0,t) = ρ(1,t) = 0. In this paper, we obtain the global existence of the weak solutions and regularity with boundary conditions (1.4)-(1.5). In order to obtain existence and higher regularity of global solutions, there are many complicated estimates on external force and higher derivations of solution to be involved, this is our 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.

The notation in this paper will be as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M14">View MathML</a> denote the usual (Sobolev) spaces on [0,1]. In addition, || · ||B denotes the norm in the space B; we also put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M15">View MathML</a>.

The rest of this paper is organized as follows. In Section 2, we shall prove the global existence in H1. In Section 3, we shall establish the global existence in H2. In Section 4, we give the detailed proof of Theorem 4.1.

2 Global existence of solutions in H1

In this section, we shall establish the global existence of solutions in H1.

Theorem 2.1 Let 0 < θ < 1, γ > 1, and assume that the initial data (ρ0,u0) satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M16">View MathML</a>and external force f satisfies f(r(x,·),·) ∈ L2n([0,T], L2n[0,1]) for some n N satisfying n(2n - 1)/(2n2 + 2n - 1) > θ, then there exists a unique global solution (ρ (x,t),u(x,t)) to problem (1.8)-(1.11), such that for any T > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M17">View MathML</a>

The proof of Theorem 2.1 can be done by a series of lemmas as follows.

Lemma 2.1 Under conditions of Theorem 2.1, the following estimates hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M18">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M19">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M20">View MathML</a>

(2.3)

where C1(T) denotes generic positive constant depending only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M21">View MathML</a>, time T and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M22">View MathML</a>.

Proof Multiplying (1.8) and (1.9) by ργ-2 and u, respectively, using integration by parts, and considering the boundary conditions (1.10), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M23">View MathML</a>

(2.4)

Integrating (2.4) with respect to t over [0,t], using Young's inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M24">View MathML</a>

which, by virtue of Gronwall's inequality and assumption f(r(x,·),·) ∈ L2n([0,T], L2n[0,1]), gives (2.1).

We derive from (1.8) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M25">View MathML</a>

(2.5)

Integrating (2.5) with respect to t over [0,t] yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M26">View MathML</a>

(2.6)

Integrating (1.9) with respect to x, applying the boundary conditions (1.10), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M27">View MathML</a>

(2.7)

Inserting (2.7) into (2.6) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M28">View MathML</a>

(2.8)

Thus, the Hölder inequality and (2.1) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M29">View MathML</a>

(2.9)

and (2.2) follows from (2.8) and (2.9).

Multiplying (1.9) by 2nu2n-1 and integrating over x and t, applying the boundary conditions (1.10), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M30">View MathML</a>

(2.10)

Applying the Young inequality and condition f(r(x, ·), ·) ∈ L2n([0,T],L2n[0,1]) to the last two terms in (2.10) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M31">View MathML</a>

(2.11)

Applying Gronwall's inequality, we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M32">View MathML</a>

(2.12)

, which, along with (2.11), yields (2.3). The proof of Lemma 2.1 is complete.

Lemma 2.2 Under conditions of Theorem 2.1, the following estimates hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M33">View MathML</a>

(2.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M34">View MathML</a>

(2.14)

Proof We derive from (2.5) and (1.9) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M35">View MathML</a>

(2.15)

Integrating it with respect to t over [0,t], we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M36">View MathML</a>

(2.16)

Multiplying (2.16) by [(ρθ) x]2n-1, and integrating the resultant with respect to x to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M37">View MathML</a>

(2.17)

here, we use the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M38">View MathML</a>. Using Young's inequality and assumptions of external of f, we get from (2.17) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M39">View MathML</a>

(30)

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M40">View MathML</a>

(2.18)

Using the Gronwall inequality to (2.18), we obtain (2.13).

The proof of (2.14) can be found in [3], please refer to Lemma 2.3 in [3] for detail.

Lemma 2.3 Under the assumptions in Theorem 2.1, for any 0 ≤ t T, we have the following estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M41">View MathML</a>

(2.19)

Proof Multiplying (1.9) by ut, then integrating over [0,1] × [0,t], we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M42">View MathML</a>

(2.20)

Using integration by parts, (1.8) and the boundary conditions (1.10), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M43">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M44">View MathML</a>

Using Lemmas 2.1-2.2, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M45">View MathML</a>

(2.21)

The last term on the right-hand side of (2.21) can be estimated as follows, using (1.8), conditions (1.10) and Lemmas 2.1-2.2,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M46">View MathML</a>

(2.22)

Inserting the above estimate into (2.21),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M47">View MathML</a>

which, by virtue of Gronwall's inequality, (2.1) and (2.14), gives (2.19).

Proof of Theorem 2.1 By Lemmas 2.1-2.3, we complete the proof of Theorem 2.1.

3 Global existence of solutions in H2

For external force f(r, t), we suppose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M48">View MathML</a>

(3.1)

Constant C2(T) denotes generic positive constant depending only on the H 2-norm of initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M49">View MathML</a>, time T and constant C1(T).

Remark 3.1 By (3.1), we easily know that assumptions (3.1) is equivalent to the following conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M50">View MathML</a>

(3.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M51">View MathML</a>

(3.3)

Therefore, the generic constant C2(T) depending only on the norm of initial data (ρ0,u0) in H2, the norms of f in the class of functions in (3.2)-(3.3) and time T.

Theorem 3.1 Let 0 < θ < 1, γ > 1, and assume that the initial data satisfies (ρ0,u0) ∈ H2 and external force f satisfies conditions (3.1), then there exists a unique global solution (ρ (x,t),u(x,t)) to problem (1.8)-(1.11), such that for any T > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M52">View MathML</a>

(3.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M53">View MathML</a>

(3.5)

The proof of Theorem 3.1 can be divided into the following several lemmas.

Lemma 3.2 Under the assumptions in Theorem 3.1, for any 0 ≤ t T, we have the following estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M54">View MathML</a>

(3.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M55">View MathML</a>

(3.7)

Proof Differentiating (1.9) with respect to t, multiplying the resulting equation by ut in L2[0,1], performing an integration by parts, and using Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M56">View MathML</a>

(3.8)

Integrating (3.8) with respect to t, applying the interpolation inequality, we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M57">View MathML</a>

(3.9)

On the other hand, by (1.9), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M58">View MathML</a>

(3.10)

We derive from assumption (3.1) and (3.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M59">View MathML</a>

(3.11)

Inserting (3.11) into (3.9), by virtue of Lemmas 2.1-2.3 and assumption (3.1), we obtain (3.6). We infer from (1.9),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M60">View MathML</a>

(3.12)

Multiplying (3.12) by uxx in L2[0,1], we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M61">View MathML</a>

(3.13)

Using Young's inequality and Sobolev's embedding theorem W1,1 W, Lemma 2.1 and (3.6), we deduce from (3.13) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M62">View MathML</a>

whence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M63">View MathML</a>

(3.14)

Applying embedding theorem, we derive from (3.14) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M64">View MathML</a>

which, along with (3.14), gives (3.7). The proof is complete.

Lemma 3.3 Under the assumptions in Theorem 3.1, for any 0 ≤ t T, we have the following estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M65">View MathML</a>

(3.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M66">View MathML</a>

(3.16)

Proof Differentiating (1.9) with respect to x, exploiting (1.8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M67">View MathML</a>

(3.17)

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M68">View MathML</a>

(3.18)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M69">View MathML</a>

Multiplying (3.18) by ρθ-1ρxx, integrating the resultant over [0,1], using condition (3.1), Young's inequality, Lemma 3.2 and Theorem 2.1, we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M70">View MathML</a>

(3.19)

Integrating (3.19) with respect to t over [0,t], using Theorem 2.1, Lemma 3.2 and the interpolation inequality, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M71">View MathML</a>

(3.20)

which, along with Lemma 2.1, gives estimate (3.15).

Differentiating (1.9) with respect to x, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M72">View MathML</a>

(3.21)

Integrating (3.21) with respect to x and t over [0,1] × [0,t], applying the embedding theorem, Lemmas 2.1-2.3 and Lemma 3.1, and the estimate (3.15), we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M73">View MathML</a>

(3.22)

The proof is complete.

Proof of Theorem 3.1 By Lemmas 3.2-3.3, Theorem 2.1 and Sobolev's embedding theorem, we complete the proof of Theorem 3.1.

4 Global existence of solutions in H4

For external force f(r,t), besides (3.1), we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M74">View MathML</a>

(4.1)

Remark 4.1 By (4.1), we easily know that assumptions (4.1) is equivalent to the following conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M75">View MathML</a>

(4.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M76">View MathML</a>

(4.3)

Therefore, the generic constant C4(T) depending only on the norm of initial data (ρ0,u0) in H4, the norms of f in the class of functions in (4.2)-(4.3) and time T.

Theorem 4.1 Let 0 < θ < 1, γ > 1, and assume that the initial data satisfies (ρ0,u0) ∈ H4 and external force f satisfies conditions (4.1), then there exists a unique global solution (ρ (x,t),u(x,t)) to problem (1.8)-(1.11), such that for any T > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M77">View MathML</a>

(4.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M78">View MathML</a>

(4.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M79">View MathML</a>

(4.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M80">View MathML</a>

(4.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M81">View MathML</a>

(4.8)

The proof of Theorem 4.1 can be divided into the following several lemmas.

Lemma 4.2 Under the assumptions of Theorem 4.1, the following estimates hold for any t ∈ [0,T],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M82">View MathML</a>

(4.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M83">View MathML</a>

(4.10)

Proof We easily infer from (1.9) and Theorem 2.1, Theorem 3.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M84">View MathML</a>

(4.11)

Differentiating (1.9) with respect to x and exploiting Lemmas 2.1-2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M85">View MathML</a>

(4.12)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M86">View MathML</a>

(4.13)

Differentiating (1.9) with respect to x twice, using Lemmas 2.1-2.3, 3.2-3.3 and the embedding theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M87">View MathML</a>

(4.14)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M88">View MathML</a>

(4.15)

Differentiating (1.9) with respect to t, and using Lemmas 2.1-2.3 and (1.8), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M89">View MathML</a>

(4.16)

which together with (4.12) and (4.14) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M90">View MathML</a>

(4.17)

Thus, estimate (4.9) follows from (4.12), (4.14), (4.17) and condition (4.1).

Now differentiating (1.9) with respect to t twice, multiplying the resulting equation by utt in L2([0,1]), and using integration by parts, (1.8) and the boundary condition (1.10), we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M91">View MathML</a>

(4.18)

here, we use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M92">View MathML</a>. Integrating (4.18) with respect to t, applying assumption (4.1) and (4.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M93">View MathML</a>

which, with Lemmas 2.1-2.3 and Theorem 3.1, implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M94">View MathML</a>

(4.19)

If we apply Gronwall's inequality to (4.19), we conclude (4.11). The proof is complete.

Lemma 4.3 Under the assumptions of Theorem 4.1, the following estimate holds for any t ∈ [0,T],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M95">View MathML</a>

(4.20)

Proof Differentiating (1.9) with respect to x and t, multiplying the resulting equation by utx in L2[0,1], and integrating by parts, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M96">View MathML</a>

(4.21)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M97">View MathML</a>

Employing Theorem 2.1, Theorem 3.1 Lemma 4.2 and the interpolation inequality, we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M98">View MathML</a>

(4.22)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M99">View MathML</a>

Applying Young's inequality several times, we have that for any ε ∈ (0,1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M100">View MathML</a>

(4.23)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M101">View MathML</a>

(4.24)

Thus we infer from (4.22)-(4.24) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M102">View MathML</a>

(4.25)

which, together with Theorem 2.1, Theorem 3.1 and Lemma 4.2, implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M103">View MathML</a>

(4.26)

On the other hand, differentiating (1.9) with respect to x and t, and using Theorem 3.1 and Lemma 4.2, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M104">View MathML</a>

(4.27)

Inserting (4.27) into (4.26), employing Theorem 2.1, Theorem 3.1 and Lemma 4.2, we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M105">View MathML</a>

(4.28)

Similarly, by Theorem 2.1, Theorem 3.1, Lemma 4.2 and the embedding theorem, we get that for any ε ∈ (0,1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M106">View MathML</a>

(4.29)

By virtue of assumption (4.1), Theorem 2.1 and Theorem 3.1, we derive that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M107">View MathML</a>

which, combined with (4.21) and (4.27)-(4.29), gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M108">View MathML</a>

(4.30)

Integrating (4.30) with respect to t, picking ε small enough, using Theorem 2.1 and Theorem 3.1, Lemma 4.2 and assumption (4.1), we complete the proof of estimate (4.20).

Lemma 4.4 Under the assumptions of Theorem 4.1, the following estimates hold for any t ∈ [0,T],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M109">View MathML</a>

(4.31)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M110">View MathML</a>

(4.32)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M111">View MathML</a>

(4.33)

Proof Differentiating (3.18) with respect to x, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M112">View MathML</a>

(4.34)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M113">View MathML</a>

(4.35)

An easy calculation with the interpolation inequality, Theorem 2.1 and Theorem 3.1, gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M114">View MathML</a>

(4.36)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M115">View MathML</a>

(4.37)

By virtue of Theorem 2.1 and Theorem 3.1, we infer from (4.36)-(4.37), (4.20) and assumption (4.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M116">View MathML</a>

(4.38)

Now multiplying (4.34) by ρθ-1ρxxx in L2[0,1], we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M117">View MathML</a>

(4.39)

Integrating (4.39) with respect to t, using Theorem 2.1 and Theorem 3.1, assumption (4.1) and (4.38), we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M118">View MathML</a>

(4.40)

By virtue of Theorem 2.1 and Theorem 3.1, we infer from (4.10), (4.15) and (4.40) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M119">View MathML</a>

(4.41)

Differentiating (1.9) with respect to t, using Theorem 2.1 and Theorem 3.1 and Lemmas 4.2-4.3, we infer that for any t ∈ [0, T],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M120">View MathML</a>

(4.42)

which, combined with (4.15), (4.40) and (4.42), gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M121">View MathML</a>

(4.43)

Differentiating (4.34) with respect to x, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M122">View MathML</a>

(4.44)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M123">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M124">View MathML</a>

Using the embedding theorem, (1.8), Theorem 2.1, Theorem 3.1 and Lemmas 4.1-4.2, we can deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M125">View MathML</a>

(4.45)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M126">View MathML</a>

(4.46)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M127">View MathML</a>

(4.47)

Inserting (4.46) into (4.47), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M128">View MathML</a>

(4.48)

By virtue of Theorems 2.1, 3.1, Lemmas 4.2-4.3, we derive from (4.40)-(4.43) and assumption (4.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M129">View MathML</a>

(4.49)

Multiplying (4.44) by ρθ-1ρxxxx in L2[0,1], we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M130">View MathML</a>

(4.50)

Integrating (4.50) with respect to t, using condition (4.1) and (4.49), we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M131">View MathML</a>

(4.51)

Differentiating (1.9) with respect to x three times, using Theorems 2.1, 3.1, Lemmas 4.2-4.3 and the interpolation inequality, we infer

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M132">View MathML</a>

(4.52)

Thus we conclude from (1.8), (4.27), (4.41), (4.43), (4.51) and assumption (4.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M133">View MathML</a>

(4.53)

Thus (4.31) follows from (4.40) and (4.51), we can derive estimate (4.32)-(4.33) from Theorem 2.1, Theorem 3.1, Lemmas 4.2-4.3, (4.41), (4,43) and (4.53). The proof is complete.

Proof of Theorem 4.1 Using (1.8),Theorem 2.1, 3.1 and Lemmas 4.2-4.4 and the proper interpolation inequality, we readily get estimate (4.4)-(4.8) and complete the proof from Theorem 4.1.

Corollary 4.5 Under assumptions of Theorem 4.1 and some suitable compatibility conditions, the global solution (ρ (x,t),u(x,t)) to problem (1.8)-(1.11) is the classical solution verifying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/43/mathml/M134">View MathML</a>

(116)

Proof By the embedding theorem, we easily prove the corollary from Theorem 4.1.

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 is in part supported by Doctoral Foundation of North China University of Water Sources and Electric Power (No. 201087), the Natural Science Foundation of Henan Province of China (No. 112300410040) and the NNSF of China (No. 11101145).

References

  1. Hoff, D, Serre, D: The failure of continuous dependence on initial data for the Navier- Stokes equations of compressible flow. SIMA J Appl Math. 51, 887–898 (1991). Publisher Full Text OpenURL

  2. Fang, D, Zhang, T: Discontinuous solutions of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. J Math Anal Appl. 318, 224–245 (2006). Publisher Full Text OpenURL

  3. Jiang, S, Xin, Z, Zhang, P: Global weak solutions to 1D compressible isentropic Navier-stokes equations with density-dependent viscosity. Meth Appl Anal. 12, 239–252 (2005)

  4. Liu, T, Xin, Z, Yang, T: Vacuum states of compressible flow. Discret Cont Dyn Syst. 4, 1–32 (1998)

  5. Okada, M, Matušů-Nečasová, Š, Makino, T: Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity. Ann Univ Ferrara Sez VII (N.S.). 48, 1–20 (2002)

  6. Xin, Z, Yao, Z: The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations.

  7. Yang, T, Yao, Z, Zhu, C: Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Commun Partial Differ Equ. 26, 965–981 (2001). Publisher Full Text OpenURL

  8. Qin, Y, Huang, L, Yao, Z: Regularity of 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity. J Differ Equ. 245, 3956–3973 (2008). Publisher Full Text OpenURL

  9. Qin, Y, Huang, L, Yao, Z: A remark on regularity of 1D compressible isentropic Navier- Stokes equations with density-dependent viscosity. J Math Anal Appl. 351, 497–508 (2009). Publisher Full Text OpenURL

  10. Yang, T, Zhao, H: A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. J Differ Equ. 184, 163–184 (2002). Publisher Full Text OpenURL

  11. Yang, T, Zhu, C: Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Commun Math Phys. 230, 329–363 (2002). Publisher Full Text OpenURL

  12. Vong, S, Yang, T, Zhu, C: Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum(II). J Differ Equ. 192, 475–501 (2003). Publisher Full Text OpenURL

  13. Fang, D, Zhang, T: Compressible Navier-Stokes equations with vacuum state in one dimension. Commun Pure Appl Anal. 3, 675–694 (2004)

  14. Fang, D, Zhang, T: A note on compressible Navier-Stokes equations with vacuum state in one dimension. Nonlinear Anal. 58, 719–731 (2004). Publisher Full Text OpenURL

  15. Qin, Y, Huang, L, Deng, S, Ma, Z, Su, X, Yang, X: Interior regularity of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Discret Cont Dyn Syst Ser S. 2, 163–192 (2009)

  16. Qin, Y, Zhao, Y: Global existence and asymptotic behavior of the compressible Navier- Stokes equations for a 1D isothermal viscous gas. Math Models Methods Appl Sci. 18, 1383–1408 (2008). Publisher Full Text OpenURL

  17. Zhang, T, Fang, D: Global behavior of compressible Navier-Stokes equations with a degenerate viscosity coefficient. Arch Ration Mech Anal. 182, 223–253 (2006). Publisher Full Text OpenURL