三维不可压Boussinesq方程组的正则性准则
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摘要
主要考虑三维不可压Boussinesq方程组的正则性准则。证明了当速度场的部分分量满足■时,局部解可以连续延拓到端点。这一结果改进和发展了三维不可压Boussinesq方程组的正则性准则,是正则性理论的一个补充。
The regularity criteria of the three-dimensional incompressible Boussinesq equations are mainly considered. It is proven that if one component of the velocity field satisfies ■to the Boussinesq equations, the local solution can be continuously extended to the endpoint. This result improves and develops some known regularity criteria of the three-dimensional incompressible Boussinesq equations, which is a supplement to the regularity theory.
The regularity criteria of the three-dimensional incompressible Boussinesq equations are mainly considered. It is proven that if one component of the velocity field satisfies ■to the Boussinesq equations, the local solution can be continuously extended to the endpoint. This result improves and develops some known regularity criteria of the three-dimensional incompressible Boussinesq equations, which is a supplement to the regularity theory.
引文
[1] GUI G,HUANG J,ZHANG P.Large global solutions to the 3D inhomogeneous Navier Stokes equations [J].J Funct Anal,2011,261:3181-3210.
[2] CHEMIN J Y,GALLAGHER I.On the global wellposedness of the 3D Navier Stokes equations with large data [J].Ann Sci Ecole Norm sup,2006,39(4) :679-698.
[3] 李率杰,李鹏,冯兆永,姚正安.基于Navier-Stokes方程的图像修复算法[J].中山大学学报(自然科学版),2012,51(1):9-13.LI S J,LI P,FENG Z Y,YAO Z A.A new algorithm for image inpainting based on the Navier-Stokes equations [J].Acta Scientiarum Naturalium Universities Sunyatseni,2012,51(1):9-13.
[4] 邹杨,冯兆永,姚正安.Navier-Stokes方程在图像放大中的应用[J].中山大学学报(自然科学版),2015,54(1):1-9.ZOU Y,FENG Z Y,YAO Z A.Application of Navier-Stokes equation in image zooming [J].Acta Scientiarum Naturalium Universities Sunyatseni,2015,54(1):1-9.
[5] ESCAURIAZA L,SEREGIN G,SVERAK V.L3,∞-solutions to the Navier-Stokes equations and backward uniqueness [J].Russian Math Surveys,2003,58:211-250.
[6] CHEMIN J Y,ZHANG P.On the critical one component regularity for 3D Navier-Stokes system [J].Ann Sci Ec Norm Super,2016,49:131-167.
[7] CHEMIN J Y,ZHANG P,ZHANG Z.On the critical one component regularity for 3D Navier-Stokes system:general case [J].Arch Ration Mech Ana,2017,224(3):871-905.
[8] HAN B,LEI Z,LI D,et al.Sharp one component regularity for Navier-Stokes [J].Arch Rational Mech Anal,2019,231(2):939-970.
[9] CHAE D.Global regularity for the 2D Boussinesq equations with partial viscosity terms [J].Adv Math,2006,203(2):497-513.
[10] CORDOBA D,FEFFERMAN C,RAFAEL D L L.On squirt singularities in hydrodynamics [J].SIAM J Math Anal,2004,36(1):204-213.
[11] HMIDI T,KERAANI S.Global well-posedness result for two-dimensional Boussinesq system with a zero diffusivity [J].Adv Diff Eqn,2007,12(4):461-480.
[12] XU X.Global regularity of solutions of 2D Boussinesq equations with fractional diffusion [J].Nonlinear Anal,2010,72(2):677-681.
[13] CHAE D,NAM H.Local existence and blow-up criterion for the Boussinesq equations [J].Proc Roy Soc Edinburgh A,1997,127(5):935-946.
[14] CHAE D,KIM S-K,NAM H-S.Local existence and blow-up criterion of Holder continuous solutions of the Boussinesq equations [J].Nagoya Math J,1999,155:55-80.
[15] GUO B.Spectral method for solving two-dimensional Newton- Boussinesq equation [J].Acta Math Appl Sin,1989,5:201-218.
[16] FAN J,ZHOU Y.A note on regularity criterion for the 3D Boussinesq system with partial viscosity [J].Appl Math Lett,2009,22:802-805.
[17] ISHIMURA N,MORIMOTO H.Remarks on the blow-up criterion for the 3D Boussinesq equations [J].Math Models Methods Appl Sci,1999,9:1323-1332.
[18] QIU H,DU Y,YAO Z A.Serrin-type blow-up criteria for three-dimensional Boussinesq equations [J].Appl Anal,2010,89(10):1603-1613.
[19] QIU H,DU Y,YAO Z A.Local existence and blow-up criterion for the generalied Boussinesq equations in Besov spaces [J].Mat.Methods Appl Sci,2013,36:86-98.
[20] 邱华,姚正安.广义Boussinesq方程的正则性准则[J].中山大学学报(自然科学版),2013,52(2):43-46.QIU H,YAO Z A.Regularity criteria for the generalized Boussinesq equations [J].Acta Scientiarum Naturalium Universities Sunyatseni,2013,52(2):43-46.
[21] LI D.On Kato-Ponce and fractional Leibniz [J].Rev Mat Iberoam,2016,DOI:10.4171/rmi/1049.
[2] CHEMIN J Y,GALLAGHER I.On the global wellposedness of the 3D Navier Stokes equations with large data [J].Ann Sci Ecole Norm sup,2006,39(4) :679-698.
[3] 李率杰,李鹏,冯兆永,姚正安.基于Navier-Stokes方程的图像修复算法[J].中山大学学报(自然科学版),2012,51(1):9-13.LI S J,LI P,FENG Z Y,YAO Z A.A new algorithm for image inpainting based on the Navier-Stokes equations [J].Acta Scientiarum Naturalium Universities Sunyatseni,2012,51(1):9-13.
[4] 邹杨,冯兆永,姚正安.Navier-Stokes方程在图像放大中的应用[J].中山大学学报(自然科学版),2015,54(1):1-9.ZOU Y,FENG Z Y,YAO Z A.Application of Navier-Stokes equation in image zooming [J].Acta Scientiarum Naturalium Universities Sunyatseni,2015,54(1):1-9.
[5] ESCAURIAZA L,SEREGIN G,SVERAK V.L3,∞-solutions to the Navier-Stokes equations and backward uniqueness [J].Russian Math Surveys,2003,58:211-250.
[6] CHEMIN J Y,ZHANG P.On the critical one component regularity for 3D Navier-Stokes system [J].Ann Sci Ec Norm Super,2016,49:131-167.
[7] CHEMIN J Y,ZHANG P,ZHANG Z.On the critical one component regularity for 3D Navier-Stokes system:general case [J].Arch Ration Mech Ana,2017,224(3):871-905.
[8] HAN B,LEI Z,LI D,et al.Sharp one component regularity for Navier-Stokes [J].Arch Rational Mech Anal,2019,231(2):939-970.
[9] CHAE D.Global regularity for the 2D Boussinesq equations with partial viscosity terms [J].Adv Math,2006,203(2):497-513.
[10] CORDOBA D,FEFFERMAN C,RAFAEL D L L.On squirt singularities in hydrodynamics [J].SIAM J Math Anal,2004,36(1):204-213.
[11] HMIDI T,KERAANI S.Global well-posedness result for two-dimensional Boussinesq system with a zero diffusivity [J].Adv Diff Eqn,2007,12(4):461-480.
[12] XU X.Global regularity of solutions of 2D Boussinesq equations with fractional diffusion [J].Nonlinear Anal,2010,72(2):677-681.
[13] CHAE D,NAM H.Local existence and blow-up criterion for the Boussinesq equations [J].Proc Roy Soc Edinburgh A,1997,127(5):935-946.
[14] CHAE D,KIM S-K,NAM H-S.Local existence and blow-up criterion of Holder continuous solutions of the Boussinesq equations [J].Nagoya Math J,1999,155:55-80.
[15] GUO B.Spectral method for solving two-dimensional Newton- Boussinesq equation [J].Acta Math Appl Sin,1989,5:201-218.
[16] FAN J,ZHOU Y.A note on regularity criterion for the 3D Boussinesq system with partial viscosity [J].Appl Math Lett,2009,22:802-805.
[17] ISHIMURA N,MORIMOTO H.Remarks on the blow-up criterion for the 3D Boussinesq equations [J].Math Models Methods Appl Sci,1999,9:1323-1332.
[18] QIU H,DU Y,YAO Z A.Serrin-type blow-up criteria for three-dimensional Boussinesq equations [J].Appl Anal,2010,89(10):1603-1613.
[19] QIU H,DU Y,YAO Z A.Local existence and blow-up criterion for the generalied Boussinesq equations in Besov spaces [J].Mat.Methods Appl Sci,2013,36:86-98.
[20] 邱华,姚正安.广义Boussinesq方程的正则性准则[J].中山大学学报(自然科学版),2013,52(2):43-46.QIU H,YAO Z A.Regularity criteria for the generalized Boussinesq equations [J].Acta Scientiarum Naturalium Universities Sunyatseni,2013,52(2):43-46.
[21] LI D.On Kato-Ponce and fractional Leibniz [J].Rev Mat Iberoam,2016,DOI:10.4171/rmi/1049.