广义Navier-Stokes方程弱解的正则性

数学物理学报 ›› 2011, Vol. 31 ›› Issue (6): 1601-1609.

广义Navier-Stokes方程弱解的正则性

边东芬, 原保全

河南理工大学数学与信息科学学院 |河南 454000

收稿日期:2010-09-15 修回日期:2011-11-10 出版日期:2011-12-25 发布日期:2011-12-25
基金资助:
国家自然科学基金(11071057)、河南省杰出青年计划(104100510015)、河南省科技创新人才计划(2009HASTIT007)和河南理工大学博士基金(B2008-62)资助

Regularity of Weak Solutions to the Generalized Navier-Stokes Equations

BIAN Dong-Fen, YUAN Bao-Quan

School of Mathematics and Informatics, Henan Polytechnic University, Henan 454000

Received:2010-09-15 Revised:2011-11-10 Online:2011-12-25 Published:2011-12-25
Supported by:
国家自然科学基金(11071057)、河南省杰出青年计划(104100510015)、河南省科技创新人才计划(2009HASTIT007)和河南理工大学博士基金(B2008-62)资助

摘要/Abstract

摘要：

该文研究象征是m(ξ)的广义Navier-Stokes方程弱解正则性. 设m(ξ)≥|ξ|^α/g(|ξ|), 其中α≥1/2+n/4, g≥1 是径向对称非减函数, 满足∫^+ ∞₁ds/sg²(s)=+∞, 如果∫^t₀||u(s)||²_B^α2, _∞ds<+∞, 那么广义Navier-Stokes 方程存在唯一整体光滑解.

关键词: 广义Navier-Stokes 方程, Besov空间, 整体光滑解

Abstract:

This paper discusses the regularity of weak solutions to the generalized Navier-Stokes equations with symbol m(ξ). Assume that m(ξ)≥|ξ|^α/g(|ξ|), where α≥1/2+n/4, g≥1, is radially symmetric, nondecreasing and satisfies ∫^+ ∞₁ds/sg²(s)=+∞, If ∫^t₀||u(s)||²_B^α2, _∞ds<+∞, then the generalized Navier-Stokes equation has a unique global smooth solution.

Key words: Generalized Navier-Stokes equations, Besov spaces, Global smooth solution

中图分类号:

35Q30

边东芬, 原保全. 广义Navier-Stokes方程弱解的正则性[J]. 数学物理学报, 2011, 31(6): 1601-1609.

BIAN Dong-Fen, YUAN Bao-Quan. Regularity of Weak Solutions to the Generalized Navier-Stokes Equations[J]. Acta mathematica scientia,Series A, 2011, 31(6): 1601-1609.

参考文献

[1] Beale J, Kato T, Majda A. Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm Math Phys, 1984, 94: 61--66

[2] He C. Regularity for solutions to the Navier-Stokes equations with one velocity component regular. Electron J Differential Equations, 2002, 2002(29): 1--13

[3] Katz N, Pavlovic N. A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Sokes equation with hyper-dissipation. Geom Funct Anal,
2002, 12: 355--379

[4] Kozono H, Taniuchi Y. Bilinear estimates in BMO and the Navier-stokes equations. Math Z, 2000, 235: 173--194

[5] Kozono H, Ogawa T, Taniuchi Y. The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math Z, 2002, 242: 251--278

[6] Kukavica I, Ziane M. One component regularity for the Navier-Stokes equations. Nonlinearity, 2006, 19: 453--469

[7] Leray J. Etude de diverses equations integrates nonlineaires et quelque problemes que pose 1'hydrodynamique. J Math Pures Appl, 1933, 12: 1--82

[8] Leray J. Essai sur les mouvements plans d'un liquide visqueux que limitent des parois. J Math Pures Appl, 1934, 13: 331--418

[9] Leray J. Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math, 1934, 63: 193--248

[10] 苗长兴.调和分析及其在偏微分方程中的应用. 北京: 科学出版社, 2004

[11] Neustupa J, Novotn\'{y} A, Penel P. An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity. Topics in Mathematical Fluid Mechanics Quad Mat, 2002, 10: 163--183

[12] Serrin J. On the interior regularity of weak solutions of the Navier-stokes equations. Arch Ration Mech Anal, 1962, 9: 187--195

[13] Serrin J. The Initial Value Problem for the Navier-Stokes Equations//R.E. Langer(ed.). Nonlinear Problems. Madison: University of Wisconsin Press, 1963: 69--98

[14] Tao T. Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equations. Anal PDE, 2009, 2(3): 361--366

[15] Triebel H. Theory of Function Spaces. Basel: Birkhäuser Verlag, 1983

[16] Beirào da Veiga H. Concerning the regularity problem for the solution of the Navier-stokes equations. C R Acad Sci Paris Sér I Math, 1995, 321: 405--408

[17] Beirào da Veiga H. A new regularity class for the Navier-Stokes equations in Rⁿ. Chinese Ann Math (Ser B), 1995, 16: 407--412

[18] Wu J. Generalized MHD equations. J Differemtial Equations, 2003, 195: 284--312

[19] Wu J. Global regularity for a class of generalized magnetohydrodynamic equations. J Math Fluid Mech, doi 10.1007/s00021-009-0017-y

[20] Zhou Y. A new regularity criterion for weak solutions to the Navier-Stokes equations. J Math Pures Appl, 2005, 84: 1496--1514

[21] Zhou Y. A new regularity result for the Navier-Stokes equations in terms of the gradient of one velocity component. Methods Appl Anal, 2002, 9: 563--578

[22] Zhou Y, Pokorn\'{y} M. On the regularity of the solutions of the Navier-Stokes equations via one velocity component. Nonlinearity, 2010, 23: 1097--1107

[23] Zhou Y, Pokorn\'{y} M. On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component. J Math Phys, 2009, 50(12): 123514

[24] Zhou Y. Regularity criteria for the generalized viscous MHD equations. Ann I H Poincar\'{e}-AN, 2007, 24: 491--505

[25] Zhou Y. A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field. Nonlinear Anal, 2010, 72: 3643--3648