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Global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation. (English) Zbl 1360.76076
Summary: The present paper is devoted to the study of the global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation. In the absence of horizontal dissipation, we establish a growth estimate on vertical component of velocity, that is, \(\sup_{p\geq 2} \frac{\| u_2(t)\|_{L^p}}{\sqrt{p\log p}}\) which is close to \(\| u_2(t)\|_{L^\infty}\) and is bounded via the low-high decomposition technique. This together with the smoothing effect in vertical direction enables us to obtain the \(H^1\)-estimate for velocity. Based on this, we prove the existence and uniqueness of classical solution without smallness assumptions. In addition, we also discuss the global well-posedness result for the rough initial data.

MSC:
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
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