Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:255

Article

Larios, Adam¹  Lunasin, Evelyn²  Titi, Edriss S.^3,4,5 

[1] Texas A&M Univ, Dept Math, College Stn, TX 77843 USA

[2] Univ Michigan, Dept Math, Ann Arbor, MI 48104 USA

[3] Univ Calif Irvine, Dept Math, Irvine, CA 92697 USA

[4] Univ Calif Irvine, Dept Mech & Aerosp Engn, Irvine, CA 92697 USA

[5] Weizmann Inst Sci, Dept Comp Sci & Appl Math, IL-76100 Rehovot, Israel

关键词: Anisotropic Boussinesq equations;    Existence;    Uniqueness;    Regularity theory;   

DOI  :  10.1016/j.jde.2013.07.011

来源: Elsevier

【 摘 要 】

We establish global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with anisotropic viscosity acting only in the horizontal direction, which arises in ocean dynamics models. Global well-posedness for this system was proven by Danchin and Paicu; however, an additional smoothness assumption on the initial density was needed to prove uniqueness. They stated that it is not clear whether uniqueness holds without this additional assumption. The present work resolves this question and we establish uniqueness without this additional assumption. Furthermore, the proof provided here is more elementary; we use only tools available in the standard theory of Sobolev spaces, and without resorting to para-product calculus. We use a new approach by defining an auxiliary stream-function associated with the density, analogous to the stream-function associated with the vorticity in 2D incompressible Euler equations, then we adapt some of the ideas of Yudovich for proving uniqueness for 2D Euler equations. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free