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 of Cao and Wu (Arch Rational Mech Anal, 208:985–1004, 2013), 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.
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Li, J., Titi, E.S. Global Well-Posedness of the 2D Boussinesq Equations with Vertical Dissipation. Arch Rational Mech Anal 220, 983–1001 (2016). https://doi.org/10.1007/s00205-015-0946-y
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DOI: https://doi.org/10.1007/s00205-015-0946-y