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The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion. (English) Zbl 1354.35087
Summary: In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized à la Leray through a smoothing kernel of order \(\alpha\) in the nonlinear term and a \(\beta\)-fractional Laplacian; we consider the critical case \(\alpha + \beta = \frac{5}{4}\) and we assume \(\frac{1}{2} < \beta < \frac{5}{4}\). The temperature equation is a pure transport equation, where the transport velocity is regularized through the same smoothing kernel of order \(\alpha\). We prove global well posedness when the initial velocity is in \(H^r\) and the initial temperature is in \(H^{r - \beta}\) for \(r > \max(2 \beta, \beta + 1)\). This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of solutions on the initial conditions.

MSC:
35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q86 PDEs in connection with geophysics
26A33 Fractional derivatives and integrals
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