Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion |
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Authors: | Adam Larios Evelyn Lunasin Edriss S. Titi |
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Affiliation: | 1. Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA;2. Department of Mathematics, University of Michigan, Ann Arbor, MI 48104, USA;3. Department of Mathematics, and Department of Mechanical and Aero-space Engineering, University of California, Irvine, Irvine, CA 92697-3875, USA;4. The Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel |
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Abstract: | 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. |
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Keywords: | 35Q35 76B03 76D03 76D09 |
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