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Global well-posedness of the 2D Boussinesq equations with vertical dissipation. (English) Zbl 1336.35297
Summary: 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 of C. Cao and J. Wu [Arch. Ration. Mech. Anal. 208, No. 3, 985–1004 (2013; Zbl 1284.35140)], 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.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids
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