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Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion. (English) Zbl 1284.35343
Summary: 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 R. Danchin and M. Paicu [Math. Models Methods Appl. Sci. 21, No. 3, 421–457 (2011; Zbl 1223.35249)]; 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 V. I. Yudovich [Zh. Vychisl. Mat. Mat. Fiz. 3, 1032–1066 (1963; Zbl 0129.19402)] for proving uniqueness for 2D Euler equations.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D09 Viscous-inviscid interaction 35Q31 Euler equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence 86A05 Hydrology, hydrography, oceanography
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