Articles

LOCAL EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS TO THE TWO DIMENSIONAL NONHOMOGENEOUS INCOMPRESSIBLE PRIMITIVE EQUATIONS

  • Quansen JIU ,
  • Fengchao WANG
Expand
  • 1. School of Mathematical Sciences, Capital Normal University, Beijing 100048, China;
    2. Advanced Institute of Natural Sciences, Beijing Normal University at Zhuhai, Zhuhai 519087, China;
    3. School of Mathematical Sciences, Beijing Normal University and Key Laboratory of Mathematics and Complex Systems, Ministry of Education Beijing 100875, China

Received date: 2019-02-15

  Revised date: 2020-05-09

  Online published: 2020-11-04

Supported by

Q.S. Jiu was partially supported by the National Natural Science Foundation of China (11671273 and 11931010), key research project of the Academy for Multidisciplinary Studies of CNU and Beijing Natural Science Foundation (1192001).

Abstract

In this article, we study the initial boundary value problem of the two-dimensional nonhomogeneous incompressible primitive equations and obtain the local existence and uniqueness of strong solutions. The initial vacuum is allowed.

Cite this article

Quansen JIU , Fengchao WANG . LOCAL EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS TO THE TWO DIMENSIONAL NONHOMOGENEOUS INCOMPRESSIBLE PRIMITIVE EQUATIONS[J]. Acta mathematica scientia, Series B, 2020 , 40(5) : 1316 -1334 . DOI: 10.1007/s10473-020-0510-1

References

[1] Bresch D, Kazhikhov A V, Lemoine J. On the two-dimensional hydrostatic Navier-Stokes equations. SIAM J Math Anal, 2004, 36:796-814
[2] Bihari I. A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math Acad Sci Hungar, 1956, 7:81-94
[3] Choe H, Kim H. Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids. Comm Partial Differential Equations, 2003, 28:1183-1201
[4] Choe H, Kim H. Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J Differential Equations, 2003, 190:504-523
[5] Cao C, Titi E S. Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Ann Math, 2007, 166:245-267
[6] Cao C, Titi E S. Global well-posedness of the 3D primitive equations with partial vertical turbulence mixing heat diffusion. Comm Math Phys, 2012, 310:537-568
[7] Cao C, Li J, Titi E S. Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity. J Differential Equations, 2014, 257(11):4108-4132
[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. Arch Ration Mech Anal, 2014, 214(1):35-76
[9] Cao C, Ibrahim S, Nakanishi K, et al. Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics. Commun Math Phys, 2015, 337(2):473-482
[10] Cao C, Li J, Titi E S. Global well-posedness of the three-dimensional primitive equations with only horizontal viscosity and diffusion. Comm Pure Appl Math, 2016, 69(8):1492-1531
[11] Cao C, Li J, Titi E S. Strong solutions to the 3D primitive equations with only horizontal dissipation:near H1 initial data. J Funct Anal, 2017, 272(11):4606-4641
[12] Ersoy M, Ngom T. Existence of a global weak solution to compressible primitive equations. C R Math Acad Sci Paris, 2012, 350:379-382
[13] Ersoy M, Ngom T, Sy M. Compressible primitive equations:formal derivation and stability of weak solutions. Nonlinearity, 2011, 24:79-96
[14] Gatapov B V, Kazhikhov A V. Existence of a global solution to one model problem of atmosphere dynamics. Sibirsk Mat Zh, 2005, 46:1011-1020
[15] Guillén-González F, Rodríguez-Bellido M A. On the strong solutions of the primitive equations in 2D domains. Nonlinear Anal Ser A:Theory Methods, 2002, 50:621-646
[16] Jiu Q, Li M, Wang F. Uniqueness of the global weak solutions to 2D compressible primitive equations. J Math Anal Appl, 2018, 461(2):1653-1671
[17] Tang T, Gao H. On the stability of weak solution for compressible primitive equations. Acta Appl Math, 2015, 140:133-145
[18] Li J, Titi E S. Existence and uniqueness of weak solutions to viscous primitive equations for a certain class of discontinuous initial data. SIAM J Math Anal, 2017, 49(1):1-28
[19] LaSalle J. Uniqueness theorems and successive approximations. Ann Math, 1949, 50:722-730
[20] Lions J L, Temam R, Wang S. New formulations for the primitive equations for the atmosphere and applications. Nonlinearity, 1992, 5(2):237-288
[21] Lions P L. Mathematical Topics in Fluid Mechanics, Vol 1:Incompressible Model. Oxford Lecture Series in Mathematic and Its Applications, Vol 3, Oxford Science Publications. New York:The Clarendon Press, Oxford University Press. 1966
[22] Pedlosky J. Geophysical Fluid Dynamics. 2nd ed. New-York:Springer-Verlag, 1987
[23] Temam R, Ziane M. Some mathematical problems in geophysical fluid dynamics//Handbook of Mathematical Fluid Dynamics. 2004
[24] Ziane M. Regularity results for Stokes type systems. Appl Anal, 1995, 58:263-292
[25] Wang F, Dou C, Jiu Q. Global Weak solutions to 3D compressible primitive equations with densitydependent viscosity. J Math Phys, 2020, 61(2):021507, 33
[26] Guo B, Huang D. On the 3D viscous primitive equations of the large-scale atmosphere. Acta Math Sci, 2009, 29B(4):846-866
Options
Outlines

/