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 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.