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On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity. (English) Zbl 1154.35073
The authors consider the 2D Boussinesq system
$\partial_t \vec v+\vec v \cdot \nabla \vec v -\nu \Delta \vec v+\nabla p=\theta \vec e,$
$\partial_t \theta+\vec v \cdot \nabla \theta -\kappa \Delta \theta=0,$
$\text{div}\;\vec v=0,$
with initial conditions
$\left. \left(\vec v,\theta\right)\right| _{t=0}=\left(\vec v^0,\theta^0\right),$
where $$\vec e=(0,1)$$, $$\vec v=(v_1,v_2)$$ is the velocity, $$p$$ is the pressure, the kinematic viscosity $$\nu$$ is a positive parameter and the diffusivity coefficient $$\kappa$$ is nonnegative. In this article, they study the specific situation where the temperature is only advected by the flow without diffusion (the molecular conductivity $$\kappa$$ is zero). They first prove provided that $$\theta^0\in L^2$$ and $$\vec v^0$$ is a divergence-free $$H^s$$-vector-field with $$s\in [0,2)$$, that system (BS) admits a global weak solution such that
$\vec v\in C({\mathbb R}_+;H^s)\cap L^2_{\text{loc}}({\mathbb R}_+;H^{\min\{s+1,2\}}),$
and
$\theta \in C_b({\mathbb R}_+; L^2).$
Moreover, if $$\theta^0\in B^0_{2,1}\cap B^0_{p,\infty}$$ with $$p\in (2,\infty]$$ and $$\vec v^0$$ is a divergence-free $$H^s$$-vector-field with $$s\in (0,2]$$, the system (BS) admits a unique global solution such that
$\vec v\in C({\mathbb R}_+;H^s)\cap L^2_{\text{loc}}({\mathbb R}_+;H^{\min\{s+1,2\}})\cap L^1_{\text{loc}}({\mathbb R}_+;B^2_{2,1}),$
and
$\theta \in C({\mathbb R}_+; B^0_{2,1}\cap B^0_{p,\infty}).$

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
##### Keywords:
Boussinesq system; zero diffusivity; global weak solution