Global well-posedness of the 2D Boussinesq equations with vertical dissipation

Abstract

We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data $(u_0,{\theta }_0)$ are required to be only in the space $X=\{f\in L^2({\mathbb{R}}^2)|{\partial }_{x}f\in L^2({\mathbb{R}}^2)\}$, and thus our result generalizes that in [C. Cao, J. Wu, Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Rational Mech. Anal., Vol. 208 (2013), 985-1004], where the initial data are assumed to be in $H^2({\mathbb{R}}^2)$. The assumption on the initial data is at the minimal level that is required to guarantee the uniqueness of the solutions. A logarithmic type limiting Sobolev embedding inequality for the $L^{\infty }({\mathbb{R}}^2)$ norm, in terms of anisotropic Sobolev norms, and a logarithmic type Gronwall inequality are established to obtain the global in time a priori estimates, which guarantee the local solution to be a global one.

References in corpus (3)

• Global Well-posedness of the 3D Primitive Equations with Only Horizontal Viscosity and Diffusion
• Global Well-posedness of an Inviscid Three-dimensional Pseudo-Hasegawa-Mima Model
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