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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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