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On the global well-posedness of a class of Boussinesq-Navier-Stokes systems. (English) Zbl 1235.76020
Summary: We consider the 2D Boussinesq-Navier-Stokes systems \begin{aligned} \partial_t u + u \cdot \nabla u + \nabla p &= - \nu |D|^\alpha u + \theta e_2 \\ \partial_t \theta+u\cdot\nabla \theta &= - \kappa|D|^\beta \theta \\ \text{div} u &= 0 \end{aligned} with $$\nu > 0$$, $$\kappa > 0$$ and $$0 < \beta < \alpha < 1$$. When $$\frac{6-\sqrt{6}}{4}({\doteq}0.888) < \alpha < 1$$ and $$1-\alpha < \beta \leq f(\alpha)$$, where $$f(\alpha ) < 1$$ is an explicit function as a technical bound, we prove global well-posedness results for the rough initial data.

##### MSC:
 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids 35B33 Critical exponents in context of PDEs 35Q35 PDEs in connection with fluid mechanics
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