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Global Well-Posedness for Certain Density-Dependent Modified-Leray- Models

Journal of Inequalities and Applications20112011:946208

Received: 3 October 2010

Accepted: 16 January 2011

Published: 13 February 2011


Global well-posedness result is established for both a 3D density-dependent modified-Leray- model and a 3D density-dependent modified-Leray- -MHD model.


Magnetic FieldMaximum PrincipleElliptic EquationRelated ModelSmooth Solution

1. Introduction

A density-dependent Leray- model can be written as

where is the fluid density, is the fluid velocity field, is the "filtered" fluid velocity, and is the pressure, which are unknowns. is the lengthscale parameter that represents the width of the filter, and for simplicity, we will take . is a bounded domain with smooth boundary .

When , the above system reduces to the well-known Leray- model and has been studied in [1, 2]. When , the above system reduces to the classical density-dependent Navier-Stokes equation, which has received many studies [36]. Specifically, it is proved in [3, 4] that the density-dependent Navier-Stokes equations has a unique locally smooth solution if the following two hypotheses (H1) and (H2) are satisfied:

(H1) for some , and in ,

(H2) and such that in .

One of the aims of this paper is to prove a global well-posedness result for the density-dependent Leray- model (1.1).

Theorem 1.1.

Let (H1) and (H2) be satisfied. Then the problem (1.1) has a unique smooth solution satisfying

for any .

Next, we consider the following density-dependent modified-Leray- -MHD model:

where and represent the unknown magnetic field and the "filtered" magnetic field, respectively. is the lengthscale parameter representing the width of the filter and we will take for simplicity. is the unit outward vector to . When and , the above system (1.3)–(1.9) reduces to the well-known density-dependent MHD equations, which have been studied by many authors (see [79] and referees therein). When and , the above system has been studied in [10] recently, and also modified models were analyzed in [11]. In this paper, we will prove the following theorem.

Theorem 1.2.

Let with , and in . Then the problem (1.3)–(1.9) has a unique smooth solution satisfying

for any .

For other related models, we refer to [1216].

Since the proof of Theorem 1.1 is similar to and simpler than that of Theorem 1.2, we only prove Theorem 1.2 for concision.

2. Proof of Theorem 1.2

By similar argument as that in [3, 4], it is easy to prove that there are and a unique smooth solution to the problem (1.3)–(1.9) in , and we only need to establish some a priori estimates for any time. Therefore, in the following estimates, we assume that the solution is sufficiently smooth.

First, it follows from (1.3), (1.7), and the maximum principle that
Testing (1.4) and (1.5) by and , respectively, using (1.3), (1.6), and (1.7), summing up them, we see that
Taking to (1.3), multiplying it by , summing over , using (1.7) and (2.3), we have
which yields
Using (1.3), (2.3) and (2.8), we find that
Multiplying (1.5) by , using (1.6), (1.7), (2.3), and (2.4), we obtain
which yields
Multiplying (1.4) by , using (1.3), (2.11), (2.12), (2.1), (2.3), and (2.4), we have
which implies
It follows from (1.4), (2.14), (2.15), (2.11), (2.12), and the -theory for Stokes system that [17]
Similarly, it follows from (1.5), (2.11), (2.12), and (2.16) that
Taking to (1.5), multiplying it by , using (1.7), (1.8), (2.12), (2.11), (2.14), and (2.15), we get
which implies
Due to (1.5), (2.3), (2.11), (2.12), (2.14), (2.19), (2.16), and the -theory of the elliptic equations, we have
Taking to (1.4), we see that
Multiplying the above equation by , using (1.3), (2.19), (2.21), (2.22), (2.9), and (2.14), we deduce that
which gives
Combining (1.4), (2.21), (2.22), (2.25), (2.14), and the regularity theory of the Stokes system [17], we obtain
Similarly, one can prove that

This completes the proof.



This work is partially supported by ZJNSF (Grant no. R6090109) and NSFC (Grant no. 10971197).

Authors’ Affiliations

College of Mathematics and Computer Science, Chongqing Three Gorges University, Wanzhou, Chongqing, China
Department of Applied Mathematics, Nanjing Forestry University, Nanjing, China
Department of Mathematics, Hokkaido University, Sapporo, Japan


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© W. Chen and J. Fan. 2011

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