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Global regularity results for the 2D Boussinesq equations with vertical dissipation. (English) Zbl 1232.35111
Authors’ abstract: This paper furthers the study of the authors [J. Differ. Equations 249, No. 5, 1078–1088 (2010; Zbl 1193.35144)] on the global regularity issue concerning the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. It is shown here that the vertical velocity \(v\) of any classical solution in the Lebesgue space \(L^q\) with \(2\leq q<\infty\) is bounded by \(C_1q\) for \(C_1\) independent of \(q\). This bound significantly improves the previous exponential bound. In addition, we prove that, if \(v\) satisfies \[ \int^T_0 \sup_{q\geq 2} {\| v(\cdot,t)\|^2_{L^q}\over q}\,dt< \infty, \] then the associated solution of the 2D Boussinesq equations preserves its smoothness on \([0, T]\). In particular, \(|v|_{L^q}\leq C_2\sqrt{q}\) implies global regularity.

MSC:
35Q30 Navier-Stokes equations
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D09 Viscous-inviscid interaction
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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