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ISRN Mathematical Analysis

Volume 2012 (2012), Article ID 796368, 7 pages

http://dx.doi.org/10.5402/2012/796368

## Regularity Criteria for Hyperbolic Navier-Stokes and Related System

^{1}Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China^{2}Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan

Received 10 July 2012; Accepted 2 August 2012

Academic Editors: S. Cingolani, P. Mironescu, L. Sanchez, and T. Tran

Copyright © 2012 Jishan Fan and Tohru Ozawa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We prove a regularity criterion for strong solutions to the hyperbolic Navier-Stokes and related equations in Besov space.

#### 1. Introduction

First, we consider the following hyperbolic Navier-Stokes equations [1]: Here is the velocity, is the pressure, and is a small relaxation parameter. We will take for simplicity.

When , (1.1) and (1.2) reduce to the standard Navier-Stokes equations. Kozono et al. [2] proved the following regularity criterion: Here is the homogeneous Besov space.

Rack and Saal [1] proved the local well posedness of the problem (1.1)–(1.3). The global regularity is still open. The first aim of this paper is to prove a regularity criterion. We will prove the following theorem.

Theorem 1.1. *Let with and in . Let be a unique strong solution to the problem (1.1)–(1.3). If satisfies
**
then the solution can be extended beyond .*

In our proof, we will use the following logarithmic Sobolev inequality [2]: and the following bilinear product and commutator estimates according to Kato and Ponce [3]: with , and .

Next, we consider the fractional Landau-Lifshitz equation: where is a three-dimensional vector representing the magnetization and is a positive constant.

When , using the standard stereographic projection , (1.9) can be rewritten as the derivative Schrödinger equation for ,

Equation (1.9) is also called the Schrödinger map and has been studied by many authors [4–31]. Guo and Han [32] proved the following regularity criterion: with .

When , Pu and Guo [33] show the local well posedness of strong solutions and the blow-up criterion with .

We will refine (1.13) as follows.

Theorem 1.2. *Let . Let be an integer such that for any . Let and and be a local smooth solution to the problem (1.9) and (1.10). If satisfies
**
for some finite , then the solution can be extended beyond .*

#### 2. Proof of Theorem 1.1

Since is a local smooth solution, we only need to prove a priori estimates.

First, testing (1.1) by and using (1.2), we see that

Testing (1.1) by and using (1.2), we find that

Applying to (1.1), testing by and using (1.2), (1.7), (1.8), and (1.6), we have

Combining (2.1), (2.2), and (2.3) and using the Gronwall inequality, we conclude that

This completes the proof.

#### 3. Proof of Theorem 1.2

Since is a local smooth solution, we only need to prove a priori estimates. In this section, we denote by the standard scalar product.

First, testing (1.9) by and using , we see that

Testing (1.9) by and using , (1.6) and (1.7), we obtain, with , which yields Here we have used the following interesting Gagliardo-Nirenberg inequalities: This completes the proof.

#### Acknowledgment

This paper is supported by NSFC (no. 11171154).

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