Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Regularity Criterion for Weak Solutions to the Navier-Stokes Equations in Terms of the Gradient of the Pressure | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 412678 6 pages doi 2008 412678 Research Article Regularity Criterion for Weak Solutions to the Navier-Stokes Equations in Terms of the Gradient of the Pressure Jishan Fan1 and Tohru Ozawa2 1 Department of Applied Mathematics Nanjing Forestry University Nanjing 210037 China 2 Department of Applied Physics Waseda University Tokyo 169-8555 Japan Correspondence should be addressed to Tohru Ozawa txozdwa@ Received 26 June 2008 Accepted 14 October 2008 Recommended by Michel Chipot We prove a regularity criterion Vn e L2 3 0 T BMO for weak solutions to the Navier-Stokes equations in three-space dimensions. This improves the available result with L2 3 0 T L . Copyright 2008 J. Fan and T. 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. 1. Introduction We study the regularity condition of weak solutions to the Navier-Stokes equations ut - ầu u-V u Vn 0 div u 0 in 0 T X R3 u t 0 u0 x x e R3. Here u is the unknown velocity vector and n is the unknown scalar pressure. For u0 e L2 R3 with div u0 0 in R3 Leray 1 constructed global weak solutions. The smoothness of Leray s weak solutions is unknown. While the existence of regular solutions is still an open problem there are many interesting sufficient conditions which guarantee that a given weak solution is smooth. A well-known condition states that if u e Lr 0 T LS R3 with - - 1 3 s x r s 2 Journal of Inequalities and Applications then the solution u is actually regular 2-8 . A similar condition m curlu e Lr 0 T Ls R3 with - - 2 s to r s 2 also implies the regularity as shown by Beirao da Veiga 9 . As regards and for s to Kozono et al. made an improvement to the following condition u e L2 0 T BTO to R3 or m e L1 0 T BTO .
