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

Abstract

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

1. Introduction

A density-dependent Leray- model can be written as

(1.1)

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 [3–6]. 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

(1.2)

for any .

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

(1.3)

(1.4)

(1.5)

(1.6)

(1.7)

(1.8)

(1.9)

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 [7–9] 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

(1.10)

for any .

For other related models, we refer to [12–16].

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

(2.1)

Testing (1.4) and (1.5) by and , respectively, using (1.3), (1.6), and (1.7), summing up them, we see that

(2.2)

Hence

(2.3)

(2.4)

(2.5)

(2.6)

Taking to (1.3), multiplying it by , summing over , using (1.7) and (2.3), we have

(2.7)

which yields

(2.8)

Using (1.3), (2.3) and (2.8), we find that

(2.9)

Multiplying (1.5) by , using (1.6), (1.7), (2.3), and (2.4), we obtain

(2.10)

which yields

(2.11)

(2.12)

Multiplying (1.4) by , using (1.3), (2.11), (2.12), (2.1), (2.3), and (2.4), we have

(2.13)

which implies

(2.14)

(2.15)

It follows from (1.4), (2.14), (2.15), (2.11), (2.12), and the -theory for Stokes system that [17]

(2.16)

Similarly, it follows from (1.5), (2.11), (2.12), and (2.16) that

(2.17)

Taking to (1.5), multiplying it by , using (1.7), (1.8), (2.12), (2.11), (2.14), and (2.15), we get

(2.18)

which implies

(2.19)

(2.20)

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

(2.21)

(2.22)

Taking to (1.4), we see that

(2.23)

Multiplying the above equation by , using (1.3), (2.19), (2.21), (2.22), (2.9), and (2.14), we deduce that

(2.24)

which gives

(2.25)

Combining (1.4), (2.21), (2.22), (2.25), (2.14), and the regularity theory of the Stokes system [17], we obtain

(2.26)

Similarly, one can prove that

(2.27)

This completes the proof.