Regularity Criteria for Hyperbolic Navier-Stokes and Related System
Jishan Fan1and Tohru Ozawa2
Academic Editor: L. Sanchez, S. Cingolani, T. Tran, P. Mironescu
Received10 Jul 2012
Accepted02 Aug 2012
Published06 Sept 2012
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)β(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)β(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 ,ββ 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 SchrΓΆdinger equation for ,
Equation (1.9) is also called the SchrΓΆdinger map and has been studied by many authors [4β31]. 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 .
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 .
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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