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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 796368, 7 pages
Regularity Criteria for Hyperbolic Navier-Stokes and Related System
1Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
2Department 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.
We prove a regularity criterion for strong solutions to the hyperbolic Navier-Stokes and related equations in Besov space.
First, we consider the following hyperbolic Navier-Stokes equations : Here is the velocity, is the pressure, and is a small relaxation parameter. We will take for simplicity.
Rack and Saal  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.
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 ,
When , Pu and Guo  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.
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
This paper is supported by NSFC (no. 11171154).
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