ISRN Mathematical AnalysisVolume 2012 (2012), Article ID 796368, 7 pagesdoi:10.5402/2012/796368

Research Article

Regularity Criteria for Hyperbolic Navier-Stokes and Related System

Jishan Fan¹ and Tohru Ozawa²

¹Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China²Department 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).

References

R. Rack and J. Saal, “Hyperbolic Navier-Stokes equations I: local well posedness,” Evolution Equations and Control Theory, vol. 1, no. 1, pp. 195–215, 2012.

H. Kozono, T. Ogawa, and Y. Taniuchi, “The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations,” Mathematische Zeitschrift, vol. 242, no. 2, pp. 251–278, 2002.

T. Kato and G. Ponce, “Commutator estimates and the Euler and Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 891–907, 1988.

I. Bejenaru, A. Ionescu, C. E. Kenig, and D. Tataru, “Equivariant Schrödinger maps in two spatial dimensions,” http://arxiv.org/abs/1112.6122

I. Bejenaru, A. D. Ionescu, and C. E. Kenig, “Global existence and uniqueness of Schrödinger maps in dimensions ,” Advances in Mathematics, vol. 215, no. 1, pp. 263–291, 2007.

I. Bejenaru, A. D. Ionescu, and C. E. Kenig, “On the stability of certain spin models in dimensions,” Journal of Geometric Analysis, vol. 21, no. 1, pp. 1–39, 2011.

I. Bejenaru, A. D. Ionescu, C. E. Kenig, and D. Tataru, “Global Schrödinger maps in dimensions : small data in the critical Sobolev spaces,” Annals of Mathematics, vol. 173, no. 3, pp. 1443–1506, 2011.

I. Bejenaru and D. Tataru, “Near soliton evolution for equivariant Schrödinger maps in two spatial dimensions,” http://arxiv.org/abs/1009.1608

I. Bejenaru, “Global results for Schrödinger maps in dimensions ,” Communications in Partial Differential Equations, vol. 33, no. 1–3, pp. 451–477, 2008.

I. Bejenaru, “On Schrödinger maps,” American Journal of Mathematics, vol. 130, no. 4, pp. 1033–1065, 2008.

N. H. Chang, J. Shatah, and K. Uhlenbeck, “Schrödinger maps,” Communications on Pure and Applied Mathematics, vol. 53, no. 5, pp. 590–602, 2000.

S. Gustafson and E. Koo, “Global well-posedness for 2D radial Schrödinger maps into the sphere,” http://arxiv.org/abs/1105.5659

S. Gustafson, K. Kang, and T. P. Tsai, “Asymptotic stability of harmonic maps under the Schrödinger flow,” Duke Mathematical Journal, vol. 145, no. 3, pp. 537–583, 2008.

S. Gustafson, K. Nakanishi, and T. P. Tsai, “Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schrödinger maps on ,” Communications in Mathematical Physics, vol. 300, no. 1, pp. 205–242, 2010.

A. D. Ionescu and C. E. Kenig, “Low-regularity Schrödinger maps,” Differential and Integral Equations, vol. 19, no. 11, pp. 1271–1300, 2006.

A. D. Ionescu and C. E. Kenig, “Low-regularity Schrödinger maps. II. Global well-posedness in dimensions ,” Communications in Mathematical Physics, vol. 271, no. 2, pp. 523–559, 2007.

J. Kato, “Existence and uniqueness of the solution to the modified Schrödinger map,” Mathematical Research Letters, vol. 12, no. 2-3, pp. 171–186, 2005.

J. Kato and H. Koch, “Uniqueness of the modified Schrödinger map in ,” Communications in Partial Differential Equations, vol. 32, no. 1–3, pp. 415–429, 2007.

C. E. Kenig, G. Ponce, and L. Vega, “On the initial value problem for the Ishimori system,” Annales Henri Poincaré, vol. 1, no. 2, pp. 341–384, 2000.

C. E. Kenig and A. R. Nahmod, “The Cauchy problem for the hyperbolic-elliptic Ishimori system and Schrödinger maps,” Nonlinearity, vol. 18, no. 5, pp. 1987–2009, 2005.

H. McGahagan, “An approximation scheme for Schrödinger maps,” Communications in Partial Differential Equations, vol. 32, no. 1–3, pp. 375–400, 2007.

F. Merle, P. Raphaël, and I. Rodnianski, “Blow up dynamics for smooth data equivariant solutions to the energy critical Schrödinger map problem,” http://arxiv.org/abs/1102.4308

F. Merle, P. Raphaël, and I. Rodnianski, “Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map,” Comptes Rendus Mathématique, vol. 349, no. 5-6, pp. 279–283, 2011.

A. Nahmod, J. Shatah, L. Vega, and C. Zeng, “Schrödinger maps and their associated frame systems,” International Mathematics Research Notices, vol. 2007, no. 21, Article ID rnm088, 29 pages, 2007.

A. Nahmod, A. Stefanov, and K. Uhlenbeck, “On Schrödinger maps,” Communications on Pure and Applied Mathematics, vol. 56, no. 1, pp. 114–151, 2003.

I. Rodnianski, Y. A. Rubinstein, and G. Staffilani, “On the global well-posedness of the one-dimensional Schrödinger map flow,” Analysis & PDE, vol. 2, no. 2, pp. 187–209, 2009.

I. Rodnianski and J. Sterbenz, “On the formation of singularities in the critical -model,” Annals of Mathematics, vol. 172, no. 1, pp. 187–242, 2010.

P. Smith, “Conditional global regularity of Schrödinger maps: sub-threshold dispersed energy,” http://arxiv.org/abs/1012.4048

P. Smith, “Global regularity of critical Schrödinger maps: subthreshold dispersedenergy,” http://arxiv.org/abs/1112.0251

T. Tao, “Gauges forthe Schrödinger map”.

T. Ozawa and J. Zhai, “Global existence of small classical solutions to nonlinear Schrödinger equations,” Annales de l'Institut Henri Poincaré, vol. 25, no. 2, pp. 303–311, 2008.

B. Guo and Y. Han, “Global regular solutions for Landau-Lifshitz equation,” Frontiers of Mathematics in China, vol. 1, no. 4, pp. 538–568, 2006.

X. Pu and B. Guo, “Well-posedness for the fractional Landau-Lifshitz equation without Gilbert damping,” Calculus of Variations and Partial Differential Equations. In press