GLOBAL WELL-POSEDNESS OF THE 2D BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION

doi:10.1007/s10473-022-0403-6

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (4): 1293-1309.doi: 10.1007/s10473-022-0403-6

GLOBAL WELL-POSEDNESS OF THE 2D BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION

晋雪婷¹, 肖跃龙², 于幻³

1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China;
2. School of Mathematics and Computational Science, Xiangtan University, Hunan, Xiangtan, 411105, China;
3. School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, China

收稿日期:2020-08-01 修回日期:2021-05-24 出版日期:2022-08-25 发布日期:2022-08-23
通讯作者: Xueting Jin,E-mail:xuetingjin@163.com E-mail:xuetingjin@163.com
基金资助:
The research is partially supported by key research grant of the Academy for Multidisciplinary Studies, CNU. Yu is partially supported by NSFC (11901040) and Beijing Municipal Commission of Education (KM202011232020) and Beijing Natural Science Foundation (1204030).

GLOBAL WELL-POSEDNESS OF THE 2D BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION

Xueting Jin¹, Yuelong Xiao², Huan Yu³

1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China;
2. School of Mathematics and Computational Science, Xiangtan University, Hunan, Xiangtan, 411105, China;
3. School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, China

Received:2020-08-01 Revised:2021-05-24 Online:2022-08-25 Published:2022-08-23
Contact: Xueting Jin,E-mail:xuetingjin@163.com E-mail:xuetingjin@163.com
Supported by:
The research is partially supported by key research grant of the Academy for Multidisciplinary Studies, CNU. Yu is partially supported by NSFC (11901040) and Beijing Municipal Commission of Education (KM202011232020) and Beijing Natural Science Foundation (1204030).

摘要/Abstract

摘要： In this paper, we prove the global well-posedness of the 2D Boussinesq equations with three kinds of partial dissipation; among these the initial data $(u_0,\theta_0)$ is required such that its own and the derivative of one of its directions $(x,y)$ are assumed to be $L^2(\mathbb R^2)$. Our results only need the lower regularity of the initial data, which ensures the uniqueness of the solutions.

关键词: Two-dimensional Boussinesq equations, global well-posedness, partial dissipation and diffusion

Abstract: In this paper, we prove the global well-posedness of the 2D Boussinesq equations with three kinds of partial dissipation; among these the initial data $(u_0,\theta_0)$ is required such that its own and the derivative of one of its directions $(x,y)$ are assumed to be $L^2(\mathbb R^2)$. Our results only need the lower regularity of the initial data, which ensures the uniqueness of the solutions.

Key words: Two-dimensional Boussinesq equations, global well-posedness, partial dissipation and diffusion

中图分类号:

35Q35

晋雪婷, 肖跃龙, 于幻. GLOBAL WELL-POSEDNESS OF THE 2D BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION[J]. 数学物理学报(英文版), 2022, 42(4): 1293-1309.

Xueting Jin, Yuelong Xiao, Huan Yu. GLOBAL WELL-POSEDNESS OF THE 2D BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION[J]. Acta mathematica scientia,Series B, 2022, 42(4): 1293-1309.

参考文献

[1] Adhikari D, Cao C, Wu J. Global regularity results for the 2D Boussinesq equations with vertical dissipation. Journal of Differential Equations, 2010, 249:1078-1088
[2] Adhikari D, Cao C, Wu J. Global regularity results for the 2D Boussinesq equations with vertical dissipation. Journal of Differential Equations, 2011, 251:1637-1655
[3] Brézis H, Gallouet T. Nonlinear Schrödinger evolution equations. Nonlinear Anal, 1980, 4(4):677-681
[4] Brézis H, Wainger S. A note on limiting cases of Sobolev embeddings and convolution inequalities. Comm Partial Differential Equations, 1980, 5(7):773-789
[5] Cannon J R, DiBenedetto E. The initial value problem for the Boussinesq equations with data in L_p//Approximation Methods for Navier-Stokes Problems. Proc Sympos, Univ Paderborn, Paderborn, 1979; Lecture Notes in Math, Vol 771. Berlin:Springer, 1980:129-144
[6] Cao C, Farhat A, Titi E S. Global well-posedness of an inviscid three-dimensional pseudo-Hasegawa-Mima model. Comm Math Phys, 2013, 319:195-229
[7] Cao C, Li J, Titi E S. Global well-posedness for the 3D primitive equations with only horizontal viscosity and diffusion. Communications in Pure and Applied Mathematics, 2016, 69(8):1492-1531
[8] Cao C, Li J, Titi E S. Local and Global Well-Posedness of Strong Solutions to the 3D Primitive Equations with Vertical Eddy Diffusivity. Archive for Rational Mechanics and Analysis, 2014, 214(1):35-76
[9] Cao C, Wu J. Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation. Arch Rational Mech Anal, 2013, 208:985-1004
[10] Chae D. Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv Math, 2006, 203:497-513
[11] Chen C, Liu J. Global well-posedness of 2D nonlinear Boussinesq equations with mixed partial viscosity and thermal diffusivity. Mathematical Methods in the Applied Sciences, 2017
[12] Constantin P, Foias C. Navier-Stokes Equations, Chicago Lectures in Mathematics. Chicago, IL:University of Chicago Press, 1988
[13] Danchin R, Paicu M. Les théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux. Bull Soc Math France, 2008, 136:261-309
[14] Danchin R, Paicu M. Global existence results for the anisotropic Boussinesq system in dimension two. Math Models Methods Appl Sci, 2011, 21:421-457
[15] Dong B, Sadek G, Chen Z. On the regularity criteria of the 3D Navier-Stokes equations in critical spaces. Acta Math Sci, 2011, 31B(2):591-600
[16] Foias C, Manley O, Temam R. Attractors for the Bénard problem:existence and physical bounds on their fractal dimension. Nonlinear Analysis, Theory, Methods and Applications, 1987, 11:939-967
[17] Hmidi T, Keraani S. On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity. Adv Differential Equations, 2007, 12:461-480
[18] Hou T, Li C. Global well-posedness of the viscous Boussinesq equations. Discrete Contin Dyn Sys, 2005, 12:1-12
[19] Hu W, Kukavica I, Ziane M. On the regularity for the Boussinesq equations in a bounded domain. J Math Phys, 2013, 54(8):081507, 10 pp
[20] Lai M, Pan R, Zhao K. Initial boundary value problem for two-dimensional viscous Boussinesq equations. Arch Ration Mech Anal, 2011, 199:739-760
[21] Larios A, Lunasin E, Titi E S. Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion. J Differential Equations, 2013, 255:2636-2654
[22] Li J, Titi E S. The 2D Boussinesq equations with vertical viscosity and vertical diffusivity. Arch Rational Mech Anal, 2016, 220:983-1001
[23] Lieb E H, Loss M. Analysis Second edition., Graduate Studies in Mathematics, 14. Providence, RI:American Mathematical Society, 2001
[24] Majda A J. Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics, Vol 9. AMS/CIMS, 2003
[25] Majda A J, Bertozzi A L. Vorticity and Incompressible Flow. Cambridge:Cambridge University Press, 2001
[26] Pedlosky J. Geophysical Fluid Dynamics. New York:Spring, 1987
[27] Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Scond Ed. Appl Math Sci, Vol 68. New York:Springer-Verlag, 1997
[28] Vallis G K. Atmospheric and Oceanic Fluid Dynamics. Cambridge Univ Press, 2006
[29] Ye Z. A note on global well-posedness of solutions to Boussinesq equations with fractional dissipation. Acta Math Sci, 2015, 35B(1):112-120

GLOBAL WELL-POSEDNESS OF THE 2D BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION

GLOBAL WELL-POSEDNESS OF THE 2D BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 10

Metrics

本文评价

推荐阅读 10

[1]	陈停停, 屈爱芳, 王振. EXISTENCE AND UNIQUENESS OF THE GLOBAL L₁ SOLUTION OF THE EULER EQUATIONS FOR CHAPLYGIN GAS[J]. 数学物理学报(英文版), 2021, 41(3): 941-958.
[2]	Muhammad Zainul ABIDIN, 陈杰诚. GLOBAL WELL-POSEDNESS FOR FRACTIONAL NAVIER-STOKES EQUATIONS IN VARIABLE EXPONENT FOURIER-BESOV-MORREY SPACES[J]. 数学物理学报(英文版), 2021, 41(1): 164-176.
[3]	苏仕斌, 赵小奎. GLOBAL WELLPOSEDNESS OF MAGNETOHYDRODYNAMICS SYSTEM WITH TEMPERATURE-DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2018, 38(3): 898-914.
[4]	张再云, 黄建华, 孙明保. ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R²[J]. 数学物理学报(英文版), 2017, 37(2): 385-394.
[5]	霍朝辉. GLOBAL WELL-POSEDNESS IN ENERGY SPACE OF SMALL AMPLITUDE SOLUTIONS FOR KLEIN-GORDON-ZAKHAROV EQUATION IN THREE SPACE DIMENSION[J]. 数学物理学报(英文版), 2016, 36(4): 1117-1152.
[6]	夏红强. GLOBAL WELL-POSEDNESS AND SCATTERING FOR THE MASS-CRITICAL HARTREE EQUATION IN HIGH DIMENSIONS[J]. 数学物理学报(英文版), 2015, 35(1): 255-274.
[7]	陈明涛. GLOBAL WELL-POSEDNESS OF THE 2D INCOMPRESSIBLE MICROPOLAR FLUID FLOWS WITH PARTIAL VISCOSITY AND ANGULAR VISCOSITY[J]. 数学物理学报(英文版), 2013, 33(4): 929-935.
[8]	蒲学科, 郭柏灵. GLOBAL WELL-POSEDNESS OF THE STOCHASTIC 2D BOUSSINESQ EQUATIONS WITH PARTIAL VISCOSITY[J]. 数学物理学报(英文版), 2011, 31(5): 1968-1984.
[9]	李用声, 杨兴雨. GLOBAL WELL-POSEDNESS FOR A FIFTH-ORDER SHALLOW WATER EQUATION ON THE CIRCLE[J]. 数学物理学报(英文版), 2011, 31(4): 1303-1317.
[10]	郭柏灵, 黄代文. ON THE 3D VISCOUS PRIMITIVE EQUATIONS OF THE LARGE-SCALE ATMOSPHERE[J]. 数学物理学报(英文版), 2009, 29(4): 846-866.