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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 796368, 7 pages
http://dx.doi.org/10.5402/2012/796368
Research Article

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.

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]: 𝜏𝑢𝑡𝑡+𝑢𝑡Δ𝑢+𝜋+𝑢𝑢+𝜏𝑢𝑡𝑢+𝜏𝑢𝑢𝑡=0,(1.1)div𝑢=0,(1.2)𝑢,𝑢𝑡𝑢(𝑥,0)=0,𝑢1(𝑥),𝑥𝑛,𝑛2.(1.3) Here 𝑢 is the velocity, 𝜋 is the pressure, and 𝜏>0 is a small relaxation parameter. We will take 𝜏=1 for simplicity.

When 𝜏=0, (1.1) and (1.2) reduce to the standard Navier-Stokes equations. Kozono et al. [2] proved the following regularity criterion: 𝜔=curl𝑢𝐿1̇𝐵0,𝑇;0,.(1.4) Here ̇𝐵0, 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 (𝑢0,𝑢1)𝐻𝑠+1×𝐻𝑠 with 𝑠>𝑛/2,𝑛2 and div𝑢0=div𝑢1=0 in 𝑛. Let (𝑢,𝜋) be a unique strong solution to the problem (1.1)–(1.3). If 𝑢 satisfies 𝑢,𝑢,𝑢𝑡𝐿1̇𝐵0,𝑇;0,,(1.5) then the solution 𝑢 can be extended beyond 𝑇>0.

In our proof, we will use the following logarithmic Sobolev inequality [2]: 𝑢𝐿𝐶1+𝑢̇𝐵0,log𝑒+𝑢𝐻𝑠(1.6) and the following bilinear product and commutator estimates according to Kato and Ponce [3]: Λ𝑠(𝑓𝑔)𝐿𝑝𝐶𝑓𝐿𝑝1Λ𝑠𝑔𝐿𝑞1+Λ𝑠𝑓𝐿𝑝2𝑔𝐿𝑞2,(1.7)Λ𝑠(𝑓𝑔)𝑓Λ𝑠𝑔𝐿𝑝𝐶𝑓𝐿𝑝1Λ𝑠1𝑔𝐿𝑞1+Λ𝑠𝑓𝐿𝑝2𝑔𝐿𝑞2,(1.8) with 𝑠>0,  Λ=(Δ)1/2 and 1/𝑝=(1/𝑝1)+(1/𝑞1)=(1/𝑝2)+(1/𝑞2).

Next, we consider the fractional Landau-Lifshitz equation: 𝜕𝑡𝜙=𝜙×Λ2𝛽𝜙𝜙,(1.9)(𝑥,0)=𝜙0(𝑥)𝕊2,𝑥𝑛,(1.10) where 𝜙𝕊2 is a three-dimensional vector representing the magnetization and 𝛽 is a positive constant.

When 𝛽=1, using the standard stereographic projection 𝕊2{}, (1.9) can be rewritten as the derivative Schrödinger equation for 𝑤, 𝑖𝑤𝑡+Δ𝑤+4(𝑤)21+|𝑤|2𝑤=0.(1.11)

Equation (1.9) is also called the Schrödinger map and has been studied by many authors [431]. Guo and Han [32] proved the following regularity criterion: 𝜙𝐿2(0,𝑇;𝐿(𝑛))(1.12) with 𝑛2.

When 0<𝛽1/2, Pu and Guo [33] show the local well posedness of strong solutions and the blow-up criterion Λ2𝛽𝜙𝐿1(0,𝑇;𝐿(𝑛))(1.13) with 𝑛3.

We will refine (1.13) as follows.

Theorem 1.2. Let 0<𝛽1/2. Let 𝑚 be an integer such that 2𝑚>(𝑛+1)/2 for any 𝑛1. Let Λ𝛽𝜙0𝐻2𝑚 and 𝜙0𝕊2 and 𝜙 be a local smooth solution to the problem (1.9) and (1.10). If 𝜙 satisfies Λ2𝛽𝜙𝐿1̇𝐵0,𝑇;0,(𝑛)(1.14) for some finite 𝑇>0, then the solution 𝜙 can be extended beyond 𝑇>0.

2. Proof of Theorem 1.1

Since (𝑢,𝜋) is a local smooth solution, we only need to prove a priori estimates.

First, testing (1.1) by 𝑢 and using (1.2), we see that 𝑑1𝑑𝑡2𝑢2+𝑢𝑢𝑡||||𝑑𝑥+𝑢2=𝑢𝑑𝑥2𝑡𝑑𝑥+𝑢𝑢𝑢𝑡u𝑑𝑥2𝑡1𝑑𝑥+2𝑢𝐿𝑢2𝐿2+𝑢𝑡2𝐿2.(2.1)

Testing (1.1) by 4𝑢𝑡 and using (1.2), we find that 𝑑𝑑𝑡2𝑢2𝑡||||+2𝑢2𝑢𝑑𝑥+42𝑡𝑑𝑥=4𝑢𝑢+𝑢𝑡𝑢𝑢𝑡𝑑𝑥𝐶𝑢𝐿𝑢2𝐿2+𝑢𝑡2𝐿2.(2.2)

Applying Λ𝑠 to (1.1), testing by Λ𝑠𝑢𝑡 and using (1.2), (1.7), (1.8), and (1.6), we have 12𝑑||Λ𝑑𝑡𝑠+1𝑢||2+||Λ𝑠𝑢𝑡||2||Λ𝑑𝑥+𝑠𝑢𝑡||2𝑑𝑥=𝑖Λ𝑠𝜕𝑖𝑢𝑖𝑢Λ𝑠𝑢𝑡Λ𝑑𝑥𝑠𝑢𝑡𝑢Λ𝑠𝑢𝑡𝑑𝑥𝑖Λ𝑠𝜕𝑖𝑢𝑖𝑢𝑡𝑢𝑖𝜕𝑖Λ𝑠𝑢𝑡Λ𝑠𝑢𝑡𝑑𝑥𝐶𝑢𝐿Λ𝑠+1𝑢𝐿2Λ𝑠𝑢𝑡𝐿2𝑢+𝐶𝑡𝐿Λ𝑠+1𝑢𝐿2+𝑢𝐿Λ𝑠𝑢𝑡𝐿2Λ𝑠𝑢𝑡𝐿2𝐶𝑢𝐿+𝑢𝐿+𝑢𝑡𝐿Λ𝑠+1𝑢2𝐿2+Λ𝑠𝑢𝑡2𝐿2𝐶1+𝑢̇𝐵0,+𝑢̇𝐵0,+𝑢𝑡̇𝐵0,log𝑒+𝑢2𝐻𝑠+1+𝑢𝑡2𝐻𝑠Λ𝑠𝑢𝑡2𝐿2+Λ𝑠+1𝑢2𝐿2.(2.3)

Combining (2.1), (2.2), and (2.3) and using the Gronwall inequality, we conclude that 𝑢𝐿(0,𝑇;𝐻𝑠+1)+𝑢𝑡𝐿(0,𝑇;𝐻𝑠)𝐶.(2.4)

This completes the proof.

3. Proof of Theorem 1.2

Since 𝜙 is a local smooth solution, we only need to prove a priori estimates. In this section, we denote by (,) the standard 𝐿2 scalar product.

First, testing (1.9) by Λ2𝛽𝜙 and using (𝑎×𝑏)𝑎=0, we see that 12𝑑||Λ𝑑𝑡𝛽𝜙||2𝑑𝑥=0.(3.1)

Testing (1.9) by Δ2𝑚Λ2𝛽𝜙 and using (𝑎×𝑏)𝑎=0, (1.6) and (1.7), we obtain, with (1/𝑝)+(1/𝑞)=(1/𝑝𝛼)+(1/𝑞𝛼)=(1/̃𝑝𝛼)+(1/̃𝑞𝛼)=1/2, 12𝑑||Δ𝑑𝑡𝑚Λ𝛽𝜙||2=𝑑𝑥𝜙×Λ2𝛽𝜙,Δ2𝑚Λ2𝛽𝜙=Δ𝑚𝜙×Λ2𝛽𝜙,Δ𝑚Λ2𝛽𝜙=Δ𝑚𝜙×Λ2𝛽𝜙+2𝑚1𝛼=1𝐶𝛼𝐷2𝑚𝛼𝜙×Λ2𝛽𝐷𝛼𝜙,Δ𝑚Λ2𝛽𝜙=Λ𝛽Δ𝑚𝜙×Λ2𝛽𝜙+2𝑚1𝛼=1𝐶𝛼𝐷2𝑚𝛼𝜙×Λ2𝛽𝐷𝛼𝜙,Δ𝑚Λ𝛽𝜙Λ𝐶2𝛽𝜙𝐿Δ𝑚Λ𝛽𝜙2𝐿2+𝐶Δ𝑚𝜙𝐿𝑝Λ3𝛽𝜙𝐿𝑞Δ𝑚Λ𝛽𝜙𝐿2+2𝑚2𝛼=1𝐶𝛼𝐷2𝑚𝛼𝜙𝐿𝑝𝛼Λ3𝛽𝐷𝛼𝜙𝐿𝑞𝛼+𝐷2𝑚𝛼Λ𝛽𝜙𝐿̃𝑝𝛼Λ2𝛽𝐷𝛼𝜙𝐿̃𝑞𝛼Δ𝑚Λ𝛽𝜙𝐿2Λ𝐶2𝛽𝜙𝐿Δ𝑚Λ𝛽𝜙2𝐿2Λ𝐶1+2𝛽𝜙̇𝐵0,Δlog𝑒+𝑚Λ𝛽𝜙𝐿2Δ𝑚Λ𝛽𝜙2𝐿2,(3.2) which yields Λ𝛽𝜙(𝑡)𝐻2𝑚𝐶.(3.3) Here we have used the following interesting Gagliardo-Nirenberg inequalities: Δ𝑚𝜙𝐿𝑝Λ𝐶2𝛽𝜙1𝜃0𝐿Δ𝑚Λ𝛽𝜙𝜃0𝐿2with𝑝=2𝑚𝛽𝑚𝛽,𝜃0=2𝑚2𝛽,Λ2𝑚𝛽3𝛽𝜙𝐿𝑞Λ𝐶2𝛽𝜙𝜃0𝐿Δ𝑚Λ𝛽𝜙1𝜃0𝐿2with𝑞=2(2𝑚𝛽)𝛽,𝐷2𝑚𝛼𝜙𝐿𝑝𝛼Λ𝐶2𝛽𝜙1𝜃𝛼𝐿Δ𝑚Λ𝛽𝜙𝜃𝛼𝐿2with𝜃𝛼=2𝑚𝛼2𝛽2𝑚𝛽,𝑝𝛼=4𝑚2𝛽,Λ2𝑚𝛼2𝛽3𝛽𝐷𝛼𝜙𝐿𝑞𝛼Λ𝐶2𝛽𝜙𝜃𝛼𝐿Δ𝑚Λ𝛽𝜙1𝜃𝛼𝐿2with𝑞𝛼=4𝑚2𝛽,𝐷𝛼+𝛽2𝑚𝛼Λ𝛽𝜙𝐿̃𝑝𝛼Λ𝐶2𝛽𝜙1̃𝜃𝛼𝐿Δ𝑚Λ𝛽𝜙̃𝜃𝛼𝐿2̃𝜃with𝛼=2𝑚𝛼𝛽2𝑚𝛽,̃𝑝𝛼=4𝑚2𝛽,Λ2𝑚𝛼𝛽2𝛽𝐷𝛼𝜙𝐿̃𝑞𝛼Λ𝐶2𝛽𝜙̃𝜃𝛼𝐿Δ𝑚Λ𝛽𝜙1̃𝜃𝛼𝐿2with̃𝑞𝛼=4𝑚2𝛽𝛼.(3.4) This completes the proof.

Acknowledgment

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

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