Open Access Research

Unique solvability of compressible micropolar viscous fluids

Mingtao Chen

Author affiliations

School of Mathematics and Statistics, Shandong University at Weihai, Weihai 264209, PR China

Citation and License

Boundary Value Problems 2012, 2012:32  doi:10.1186/1687-2770-2012-32


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/32


Received:4 December 2011
Accepted:20 March 2012
Published:20 March 2012

© 2012 Chen; 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 article, we consider the compressible micropolar viscous flow in a bounded or unbounded domain Ω ⊆ ℝ3. We prove the existence of unique local strong solutions for large initial data satisfying some compatibility conditions. The key point here is that the initial density need not be positive and may vanish in an open set.

1 Introduction

Compressible micropolar viscous fluids study the viscous compressible fluids with randomly oriented (or spherical) particles suspended in the medium, when the deformation of fluid particles is ignored. The theory can help us consider some physical phenomena which cannot be treated by the compressible viscous baratropic flows, due to the effect of microparticles. The microstructure of the polar fluids is mechanically significant. The governing system of equations of compressible micropolar viscous fluids expresses the balance of mass, momentum, and moment of momentum see [1,2], that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M1">View MathML</a>

(1)

Here the density ρ = ρ(t, x), the velocity u = [u1(t, x), u2(t, x), u3(t, x)], the microrotational velocity w = [w1(t, x), w2(t, x), w3(t, x)], and the pressure p(ρ) = γ(a > 0, γ > 1) are functions of the time t ∈ (0, T) and the spatial coordinate x , where is either a bounded open subset in ℝ3 with smooth boundary or a usual unbounded domain such as the whole space ℝ3, the half space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M2">View MathML</a> and an exterior domain with smooth boundary. μ, λ, μ', λ', and ζ are positive constants, which describe the viscosity of the fluids, satisfying the additional condition: μ + λ - ζ ≥ 0.

We prescribe the initial boundary value conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M3">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M4">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M5">View MathML</a>

(4)

There are lots of literature on the well-posedness and ill-posedness of the incompressible micropolar viscous fluid, i.e., the system (1) with ρ ≡ Constant. Yamaguchi [3] considered the global strong solution in a bounded domain with small initial data. Lukaszewicz [4] showed the existence theorem for the incompressible micropolar viscous fluid with sufficiently regular initial data. And with the effect of Magnetic fields, the system of magneto-micropolar fluid, in [5], Yuan gave the local smooth solution without the smallness of the initial data; and also gave the blow-up criterion for the smooth solutions. Amirat and Hamdache [6] considered the global weak solutions with finite energy and establish the long-time behavior of the solution. And Amirat et al. [7] also proved the global in time weak solution with finite energy of an initial-boundary value problem and long-time behavior of such solution with the effect of magnetic field. We also refer the reader to [8] for local strong solution in bounded domain of ℝ3 and references therein.

For the compressible case, in the absence of vacuum, there also have been lots of studies on the full viscous compressible micropolar fluids (which include also the conservation law of energy) since Eringen [1]. The one-dimensional problem studied by Mujaković [9-12], and Dražić and Mujaković [13] and references therein. For the multidimensional case, we refer the readers to [2,14-17] and references therein. If the vacuum occur initially, Chen [18,19] studied the global strong solutions of compressible micropolar viscous fluids in 1-D. Recently, Amirat and Hamdache [20] studied the micropolar fluids with the effect of magnetic field, they prove the global weak solution in a bounded domain in ℝ3 with initial vacuum.

Classical compressible viscous flows, i.e., w = 0 and ζ = 0 in (1), have been studied by many authors. In [21], Matsumura and Nishida considered the global smooth solutions under the condition that initial data close to a non-vacuum equilibrium. For the arbitrary initial data, the major breakthrough is due to Lions [22], where he established global existence of weak solutions for the whole space, periodic domain or bounded domains with Dirichlet boundary conditions if γ ≥ 9/5. The restriction on γ was improved to γ > 3/2 by Feireisl etal. in [23].

In [24], Choe and Kim established local in time strong solution of isentropic compressible fluids with initial density ρ0 may vanish in an open subset. After that, Cho and Kim [25] studied the local existence of strong solutions of viscous polytropic fluids with vacuum. Cho et al. [26] considered the unique solvability of the initial boundary value problems for compressible viscous fluids that the initial density need not be bounded away from zero.

Hoff and Serre [27] presented an example which showed the failure of continuous dependence of weak solution, so it should be noticed that contrary to the case of strong solutions, weak solutions with vacuum need not depend continuously on their initial data.

Before stating the main result, we introduce the notions used throughout this article. We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M6">View MathML</a>

For 1 ≤ r ≤ ∞, we denote the standard homogeneous and inhomogeneous Sobolev spaces as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M7">View MathML</a>

In spirit of [26], our aim of this article is to establish the local strong solution of (1)-(4). Assume the initial data (ρ0, w0, u0) satisfying the regularity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M8">View MathML</a>

(5)

where 3 < q < ∞, and the compatibilities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M9">View MathML</a>

(6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M10">View MathML</a>

(7)

for some (g1, g2) ∈ L2.

Now, we state our main result in this article:

Theorem 1 Assume the data (ρ0, u0, w0) satisfy the regularity conditions (5) and the compatibility conditions (6) and (7).

Then there exists a time T*∈ (0, T) and a unique strong solutions (ρ, u, w) to the initial boundary value problem (1)-(4) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M11">View MathML</a>

where q0 = min{6, q}.

The article is organized as follow. In Section 2, we prove the global existence and regularity of the unique strong solutions to a linearized problem of the nonlinear problem (1)-(4). The result is used in Section 3 to construct approximate solutions to the original nonlinear problem. In Section 3, we derive some uniform bounds of the higher derivatives independent of the lower bound of the density. Moreover, we prove the convergence and obtain the local existence of strong solutions. In Section 4, we finish the proof of Theorem 1.

Remark 1 In this article, we use the following fact frequently later:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M12">View MathML</a>

under the boundary condition of (3) or (4).

2 Global existence for the linearized equations

In this section, we consider the following linearized system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M13">View MathML</a>

(8)

where v is a known vector fields. If the initial density ρ0 is bounded away from zero, then we can apply standard arguments to prove the global existence of a unique strong solution to the initial boundary value problem (2)-(4) and (8), since the system can be uncoupled into a linear transport equation and two linear parabolic equations.

In this section, we prove the following existence result for the general case of nonnegative initial densities.

Theorem 2 Assume that the data (ρ0, u0, w0) satisfies the regularity conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M14">View MathML</a>

(9)

for some 3 < q < ∞ and the compatibility conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M15">View MathML</a>

(10)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M16">View MathML</a>

(11)

for some (g1, g2) ∈ L2. If in addition, v satisfies the regularity conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M17">View MathML</a>

where q0 = min{6, q}. Then there exists a unique strong solution (ρ, u, w) to the initial boundary value problems (2)-(4) and (8) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M18">View MathML</a>

(12)

Here, we emphasize that we focus on the bounded open domain with smooth boundary condition. As for the unbounded domain, we can deal the same problem with the standard domain expansion technique that derived in [22]. We also refer the reader to [26]. The key of this technique is that the a priori estimates do not depend on the size of the domain. So, we here emphasize that the a priori estimates deduced in this section are independent of the size of the domain.

2.1 Existence of theorem 2

We begin with an existence result for the case of positive initial densities.

Lemma 1 Let Ω be a bounded domain in 3 with smooth boundary, and let (ρ0, u0, w0) be a given data satisfying the regularity condition (9)-(11). Assume further that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M20">View MathML</a>, ρ0 H2 and ρ0 ≥ δ in Ω for some constant δ > 0. Then there exists a unique strong solution (ρ, u, w) to the initial boundary value problems (2), (3) and (8) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M21">View MathML</a>

(13)

Proof Due to the classical embedding result, that υ C([0, T];H2). Hence the existence and regularity of a unique solution of the linearized continuity Equation (8)1 have been well-known. Moreover, the unique solution ρ can be expressed by:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M22">View MathML</a>

(14)

where U(t, s, x) is the solution to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M23">View MathML</a>

We refer the readers to [28,29] for a detailed proof. As a consequence of (14) and Sobolev inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M24">View MathML</a>

(15)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M25">View MathML</a>. Hence the linearized moment of momentum Equation (8)2 can be written as a linear parabolic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M26">View MathML</a>

The existence and regularity of the unique solution w can be proved by applying classical methods, for instance, the method of continuity (see [29]). Similarly, the linearized momentum Equation (8)3 also as a linear parabolic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M27">View MathML</a>

and can be solved using the same method.

Now we prove the existence of strong solutions. Then thanks to Lemma 1, there exists a unique strong solution (ρ, u, w) satisfying the regularity (13). To remove the additional hypotheses in Lemma 1, we will derive some uniform estimates independent of δ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M28">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M29">View MathML</a>.

First, we consider the solution ρ of the linearized continuity Equation (8)1. Since (8)1 is a linear transport equation, so we need to prove the estimates. Multiply (8)1 by ρr-1(r = 2 or q0) and integrating (by parts) over , we obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M30">View MathML</a>

Then, using Sobolev inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M31">View MathML</a>

(16)

Differentiating (8)1 with respect to xi, then multiplying the resultant equation by iρ|∂iρ|r-2, i = 1, 2, 3 and then integrating over , we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M32">View MathML</a>

By virtue of Sobolev inequality, we get:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M33">View MathML</a>

(17)

Using (16) and (17) together with Gronwall's inequality, one yields:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M34">View MathML</a>

(18)

Since ρt = -υ ·∇ρ - ρ div υ, p = p(ρ) and p(0) = 0, we can easily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M35">View MathML</a>

Then, we consider the solution w of the linearized Equation (8)2. Rewrite (8)2, with the help of (8)1, as:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M36">View MathML</a>

(19)

Multiplying this equation by wt and integrating (by parts) over , we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M37">View MathML</a>

(20)

Then, using Young's inequality, we obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M38">View MathML</a>

(21)

Integrating the above inequality over (0, t), we get:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M39">View MathML</a>

(22)

Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M40">View MathML</a>

Therefore, in view of Gronwall's inequality, we have:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M41">View MathML</a>

(23)

To derive higher regularity estimates, we differentiate (19) with respect to t and obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M42">View MathML</a>

Multiplying the above equation by wt, and integrating over , we have:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M43">View MathML</a>

(24)

Now, we estimate each term on the right-hand side of (24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M44">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M45">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M46">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M47">View MathML</a>

Choosing ε= μ'/6, substitute the above estimates into (24), we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M48">View MathML</a>

(25)

Integrating over (τ, t) for some fixed τ ∈ (0, T), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M49">View MathML</a>

(26)

To estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M50">View MathML</a>, due to(19), we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M51">View MathML</a>

and thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M52">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M258">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M53">View MathML</a>

Therefore, letting τ → 0 in (26), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M54">View MathML</a>

(27)

To obtain further estimates, observe that since for each t ∈ [0, T], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M55">View MathML</a> is a solution of the elliptic system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M56">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M57">View MathML</a>. And then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M58">View MathML</a>

(28)

Therefore, using the previous estimates, we can deduce from (28) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M59">View MathML</a>

(29)

Now, we consider the solution u of the linearized Equation (8)3. Rewrite (8)3 as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M60">View MathML</a>

(30)

Multiplying this equation by ut and integrating over , we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M61">View MathML</a>

Then, using Young's inequality, we obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M62">View MathML</a>

Integrating the above inequality over (0, t) using Young's inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M63">View MathML</a>

Therefore, in view of Gronwall's inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M64">View MathML</a>

(31)

To derive higher regularity estimates, differentiate (30) with respect to t and obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M65">View MathML</a>

Multiplying the above equation by ut, integrating (by parts) over , and using (8)1, we obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M66">View MathML</a>

(32)

Using Sobolev inequality, interpolation inequality and Young's inequality, we obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M67">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M68">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M69">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M70">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M71">View MathML</a>

Substituting the above estimates into (32), we can easily show that:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M72">View MathML</a>

(33)

Now, for fixed τ ∈ (0, T). Since the right-hand side of (33) is integrable in (0, T), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M73">View MathML</a>

(34)

To estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M74">View MathML</a> from (8)3, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M75">View MathML</a>

and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M76">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M259">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M77">View MathML</a>

Therefore, letting τ → 0 in (34), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M78">View MathML</a>

(35)

To obtain further estimates, observe that for each t ∈ [0, T], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M79">View MathML</a> is a solution of the following elliptic system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M80">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M81">View MathML</a>. It follows from the elliptic regularity theory, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M82">View MathML</a>

(36)

Therefore, using the previous estimates, we get from (36) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M83">View MathML</a>

(37)

Since the estimates we have derived, we prove the existence result. First, we consider for the case of bounded domain. Using standard regularization techniques, we choose pδ = pδ(⋅) and υδ, 0 < δ ≪ 1, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M84">View MathML</a>

Then, for each δ ∈ (0, 1), let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M85">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M86">View MathML</a> is the solution to the boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M87">View MathML</a>

It follows from the elliptic regularity result that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M88">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M89">View MathML</a> as δ → 0. Hence, if we denote by (ρδ, uδ, wδ) the solution of (8) with the initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M90">View MathML</a> and (p, v) replaced by (pδ, vδ), it satisfies the estimates (35), (37), (31), (23), (18), (27), (29), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M92">View MathML</a>. Therefore, we conclude that a subsequence of solutions (ρδ, uδ, wδ) converges to a limit (ρ, u, w) in a weak sense. Then it can easy to show that (ρ, u, w) is a weak solution to the original problem (8). Moreover, due to the lower semi-continuity of various norms, we have the following regularity estimates for (ρ, u, w):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M93">View MathML</a>

(38)

2.2 Continuity and uniqueness

Now, turn our attention to the continuity of the solution (ρ, u, w). The continuity of ρ can be proved by a standard argument from the theory of hyperbolic equations. Since ρ satisfies the regularity (38), it follows from a result of DiPerna and Lions [30] and classical embedding results that (See [31]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M94">View MathML</a>

To show the strong continuity in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M95">View MathML</a>, observe from (17), (18) and i = 1, 2, 3,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M96">View MathML</a>

and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M97">View MathML</a>. Hence it follows from a well-known criterion on the strong conver-gence for the space Lr (See [32]) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M98">View MathML</a>

Therefore, the continuity of ∇ρ in Lr(r = 2, q0) follows from the result and the observation that for each fixed t0 ∈ [0, t], the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M99">View MathML</a> is a unique strong solution to the similar initial value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M100">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M101">View MathML</a>.

To show the continuity of w, we first observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M102">View MathML</a>

We now prove the continuity of ρwt in L2. For a.e. t ∈ (0, T) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M103">View MathML</a> the from (19), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M104">View MathML</a>

and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M105">View MathML</a>

(39)

Using the regularity (38) of (ρ, w), we show that the right-hand side of (39) is bounded above by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M106">View MathML</a> for some positive function A1(t) ∈ L2(0, T). Hence it follows, from the well-known result (see [31]) that (ρwt)tL2(0, T; H-1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M107">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M103">View MathML</a> where 〈⋅, ⋅〉 denotes the dual pairing of H-1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M108">View MathML</a>. Then since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M109">View MathML</a>, it follows from a standard embedding result ρwt C([0, T]; L2). Therefore, we conclude that for each t∈[0, T], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M110">View MathML</a> is a solution of the elliptic system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M111">View MathML</a>

where G1 = -ρwt-4ζw C([0, T]; L2). Now we turn to show that w C([0, T]; D2). In view of the elliptic regularity estimate (36), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M112">View MathML</a>

(40)

Using the estimate (36), we obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M113">View MathML</a>

(41)

Substituting this into (40), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M114">View MathML</a>

The continuity of ρut in L2 is similar as the proof of ρwt. From (30), for a.e. t ∈ (0, T), and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M115">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M116">View MathML</a>

and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M117">View MathML</a>

(42)

Using the regularity (38) of (ρ, u, w), we show the right-hand side of (42) is bounded above by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M118">View MathML</a> for some positive function A2(t) ∈ L2(0, T). Hence by the similar argument of ρwt, we see that (ρut)tL2(0, T; H-1), and then ρut C([0, T]; L2). Therefore, we conclude that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M119">View MathML</a> is a solution of the elliptic system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M120">View MathML</a>

where G2 = -ρut - ∇p(ρ) + 2ζ rot w C([0, T]; L2).

Now, we will show that u C([0, T]; D2). In view of the elliptic regularity estimate (36), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M121">View MathML</a>

(43)

To estimate the third term in the right-hand side of (43), using the estimate (38), similarly as (41), we get:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M122">View MathML</a>

Substituting this into (43), we conclude the continuity of u in D2. This completes the proof of the continuity.

Finally, we prove the uniqueness of solutions satisfying the regularity (38). Let (ρ1, u1, w1) and (ρ2, u2, w2) be two strong solutions to the problem (8) and (2)-(4). Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M123">View MathML</a>

Then, it follows from (8)1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M124">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M126">View MathML</a>, we conclude from Gronwall's inequality that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M127">View MathML</a>, i.e., ρ1 = ρ2 in (0, T) × . Next, we choose a cut-off function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M128">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M129">View MathML</a>

Define φR(x) = φ(x/R) for x ∈ ℝ3. From (8)2 and using the uniqueness of ρ, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M131">View MathML</a>

Then multiplying the above equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M132">View MathML</a>, integrating over (0, T) × , and letting R → ∞, we easily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M133">View MathML</a>

Hence, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M135">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M136">View MathML</a> in (0, T), due to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M137">View MathML</a>. Therefore, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M138">View MathML</a> in (0, T) × .

Similarly, from the uniqueness of ρ, and (8)3, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M139">View MathML</a>

Then multiplying it by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M140">View MathML</a> integrating over (0, T) × , and letting R → ∞, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M141">View MathML</a>

Due to ū(0) = 0, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M142">View MathML</a> in (0, T). Then ū = 0 in (0, T) × .

This completes the proof of the Theorem 2.

3 A local existence result for positive densities

In this section, we assume also that is a bounded domain in ℝ3 with smooth boundary and prove a local (in time) existence result on strong solutions with positive densities to the original nonlinear problem (1)-(4).

Proposition 1 Assume that p = γ (a > 0, γ > 1), and the data (ρ0, u0, w0) satisfies the regularity conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M144">View MathML</a>

for some q with 3 < q < ∞ and compatibilities (6)-(7). Assume further that ρ0 ≥ δ in Ω for some constant δ > 0. Then there exist a time T* ∈ (0, T) and a unique strong solution (ρ, u, w) to the nonlinear problem (1)-(3) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M145">View MathML</a>

where q0 = min(6, q). Furthermore, we have the following estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M146">View MathML</a>

(44)

The constant C and the local time T* in (44) are independent of δ.

To prove the proposition, we first construct approximate solutions, inductively, as follows:

- first define u0 = 0, and

- assume that uk-1 was defined for k ≥ 1, let (ρk, uk, wk) be the unique solution to the following initial boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M147">View MathML</a>

(45)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M148">View MathML</a>

(46)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M149">View MathML</a>

(47)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M150">View MathML</a>

(48)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M151">View MathML</a>

(49)

The existence of a global strong solution (ρk, uk, wk) with the regularity (12) to the linearized problem (45)-(49) was proved in the previous section.

From now on, we derive uniform bounds on the approximate solutions and then prove the convergence of the approximate solutions to a strong solutions of the original nonlinear problem.

3.1 Uniform bounds

Let K ≥ 1 be a fixed large integer, and let us introduce an auxiliary function ΦK(t), defined by:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M152">View MathML</a>

Then we estimate each term of ΦK in terms of some integrals of ΦK, apply arguments of Gronwall-type and thus prove that ΦK is locally bounded. First, notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M153">View MathML</a>

Then, we estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M154">View MathML</a> by ΦK.

• Estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M155">View MathML</a>

Observe that for any t ∈ [0, T], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M156">View MathML</a> is a solution of the elliptic system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M157">View MathML</a>

Hence we deduce from the elliptic regularity result that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M158">View MathML</a>

Thus we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M159">View MathML</a>

(50)

• Estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M160">View MathML</a>

Similar as the estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M155">View MathML</a>. We see that for any t ∈ [0, T], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M161">View MathML</a> is a solution of the elliptic system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M162">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M163">View MathML</a>

Thus we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M164">View MathML</a>

(51)

• Estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M165">View MathML</a>

Multiplying (46) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M166">View MathML</a>, and integrating over , using Young's inequality we obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M167">View MathML</a>

Integrating the above inequality we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M168">View MathML</a>

(52)

To estimate the right-hand side of (52), we first observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M169">View MathML</a>, together with the estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M160">View MathML</a> we conclude that:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M170">View MathML</a>

(53)

• Estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M171">View MathML</a>

Differentiating (46) with respect to t, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M172">View MathML</a>

Multiplying this by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M166">View MathML</a> and integrating over , we obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M173">View MathML</a>

(54)

Using Sobolev inequality and Young's inequality (with ε) repeatedly, we have:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M174">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M175">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M176">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M177">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M178">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M179">View MathML</a>

Substituting all these estimates into (54) and choosing ε, η > 0 small enough, we obtain that:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M180">View MathML</a>

(55)

Integrating over (τ, t) for fixed τ > 0, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M181">View MathML</a>

(56)

To estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M182">View MathML</a> as τ → 0, we multiplying (46) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M166">View MathML</a>, integrate over , we have:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M183">View MathML</a>

Using Young's inequality, together with the compatibility condition (7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M184">View MathML</a>

Substituting the above estimate into (56), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M185">View MathML</a>

(57)

• Estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M186">View MathML</a>

Multiplying (47) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M187">View MathML</a> and integrating over , we obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M188">View MathML</a>

(58)

Using (45), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M189">View MathML</a>

(59)

Integrating (58) over (0, t), using (59) and Young's inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M190">View MathML</a>

(60)

To estimate the right-hand side of (60), with the help of Sobolev inequality and Young's inequality (with ε), we have:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M191">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M192">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M193">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M194">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M130">View MathML</a>

Substitute all the above estimates into (60), we obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M195">View MathML</a>

(61)

• Estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M196">View MathML</a>

Differentiating (47) with respect to t, we obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M197">View MathML</a>

(62)

Multiplying the equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M198">View MathML</a>, and integrating over , we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M199">View MathML</a>

(63)

Using Sobolev inequality and Young's inequality we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M200">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M201">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M202">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M203">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M204">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M205">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M206">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M207">View MathML</a>

Substituting the above estimates into (63), and choosing ε small enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M208">View MathML</a>

For a fixed τ ∈ (0, T), integrating the above inequality over (τ, t), we have:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M209">View MathML</a>

(64)

To estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M210">View MathML</a> as τ → 0, we multiply (47) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M198">View MathML</a>, and integrate over . Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M211">View MathML</a>

Hence, with the help of compatibility condition (6), we get:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M212">View MathML</a>

Considering this and letting τ → 0 in (64), we finally have:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M213">View MathML</a>

(65)

Thanks to the estimate (65), (61), (57), (53), we get:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M214">View MathML</a>

If η is sufficient small such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M215">View MathML</a>, then from the recursive relation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M216">View MathML</a>, it follows from that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M217">View MathML</a>

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M218">View MathML</a>

(66)

Finally, we recall from (18) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M219">View MathML</a>

(67)

for all k, 1 ≤ k ≤ K. To estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M220">View MathML</a> for 1 ≤ k ≤ K, we invoke the elliptic regularity result (36) and the estimate (51). If 3 < q0 < 6, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M221">View MathML</a>

for some θ1, θ2 ∈ (0, 1). Thus, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M222">View MathML</a>

(68)

Similarly, from (28) and the estimate (50), we can deduce that,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M223">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M224">View MathML</a>

for some θ3, θ4 ∈ (0, 1). Thus, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M225">View MathML</a>

(69)

Thanks to the estimates (68), (69), (67), and (66), we can easily show that there exists a small time T1 ∈ (0, T) depending only on the parameters C such that the following uniform estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M226">View MathML</a>

(70)

for all k ≥ 1.

3.2 Convergence

We show that the approximate solutions converge to a solution to the original problem (1)-(4) in a strong sense. To prove this, we define:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M227">View MathML</a>

Then it follows from (45)-(47), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M228">View MathML</a>

(71)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M229">View MathML</a>

(72)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M230">View MathML</a>

(73)

Multiplying (72) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M231">View MathML</a>, integrating over , using (45) and Young's inequality we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M232">View MathML</a>

And thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M233">View MathML</a>

(74)

Multiplying (73) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M234">View MathML</a>, integrating over , using (45) and Young's inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M235">View MathML</a>

And thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M236">View MathML</a>

(75)

On the other hand, observing that (71), we can easily prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M237">View MathML</a>. Hence, multiplying (71) by sgn<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M238">View MathML</a> and integrating over , we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M239">View MathML</a>

Multiplying the above inequality by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M240">View MathML</a> on both side, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M241">View MathML</a>

(76)

Similarly, we get:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M242">View MathML</a>

(77)

Combining (74)-(77), we obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M243">View MathML</a>

(78)

for some function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M244">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M245">View MathML</a> for all 0 ≤ t ≤ T1 and k ≥ 1.

Let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M246">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M247">View MathML</a>

Then integrating (78) over (0, t) ⊂ (0, T1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M248">View MathML</a>

which implies, by virtue of Gronwall's inequality, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M249">View MathML</a>

Hence, choosing ε > 0 and then T* > 0 so small that 4(T* + ε)C1 < 1, T* < T1 and exp(C2T*) < 2, we also deduce from Gronwall's inequality that for all K ≥ 1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M250">View MathML</a>

Therefore, we conclude that (ρk, uk, wk) converges to a limit (ρ, u, w) in the following strong sense:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M251">View MathML</a>

Now it is simple to check that (ρ, u, w) is a weak solution to the original problem (1)-(4). Then, by virtue of the lower semi-continuity of norms, we deduce from the uniform bound (70) that (ρ, u, w) satisfies the following regularity estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M252">View MathML</a>

(79)

The time-continuity of the solution (ρ, u, w) can be proved by the same argument as in Section 2. This completes the proof of Proposition 3.1.

Here we should emphasize that the constant C and the local existence time T* in (79) do not depend on δ and the size of .

4 Proof of theorem 1.1

In this section, we complete the proof of Theorem 1.1.

Assume for the moment that is a bounded domain with smooth boundary. Let (ρ0, u0, w0) be the given data satisfying (5). For each small δ > 0, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M253">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M254">View MathML</a> be the unique smooth solution to the elliptic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M255">View MathML</a>

Then by virtue of Proposition 3.1, there exist a time T* ∈ (0, T) and a unique strong solution (ρδ, uδ, wδ) in [0, T*] × to the problem (1)-(4) with the initial data replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M256">View MathML</a>. Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/32/mathml/M257">View MathML</a> in H2 as δ → 0, (ρδ, uδ, wδ) satisfies the bound (44), and the constant T*, C are independent of δ. Hence, following the same argument as in the proof of Theorem 2.1, we prove the existence and regularity of a strong solution to the original problem (1) - (4). Moreover, since the constant C and the local existence T* in (44) are independent of the size of the domain, we also obtain the same existence and regularity results for unbounded domains by means of the domain expansion technique. Finally, the uniqueness can be proved by using the similar methods to the proof of the convergence in Section 3.

This completes the proof of Theorem 1.1.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author was indebted to the referee for giving valuable suggestions which improve the presentation of the article.

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