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Global Well-Posedness of the 2D Boussinesq Equations with Vertical Dissipation


We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data (Formula presented.) are required to be only in the space (Formula presented.) , 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 (Formula presented.). 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 (Formula presented.) 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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