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On the global well-posedness for Boussinesq system. (English) Zbl 1111.35032
The 2D Cauchy problem to the Boussinesq system is studied in the paper
\[ \begin{aligned} &\frac{\partial v}{\partial t}-\Delta v+(v\cdot\nabla)v+\nabla p=\theta(0,1), \quad x\in \mathbb R^2,\quad t>0,\\ &\frac{\partial \theta}{\partial t}+v\cdot\nabla \theta=0, \quad x\in \mathbb R^2,\quad t>0, \\ &\text{div}\,(v)=0, \quad v| _{t=0}=v_0(x),\quad \theta| _{t=0}=\theta_0(x),\quad x\in \mathbb R^2. \end{aligned}\tag{1} \] Here \(v=(v_1,v_2)\) is the velocity, \(p\) is the pressure and \(\theta\) is the temperature.
It is proved that problem (1) has a unique global solution \((v,p,\theta)\) if \(v_0\in L_2\cap B^{-1}_{\infty,1}\) and \(\theta_0\in B^{0}_{2,1}\). The important parts of the proof are estimates to solutions of the Stokes system
\[ \begin{aligned} &\frac{\partial u}{\partial t}-\Delta u+(v\cdot\nabla)u+\nabla p=f, \quad \text{div}\,(v)=0, \quad x\in \mathbb R^2,\quad t>0,\\ &u| _{t=0}=u_0(x) \end{aligned} \] . and of the transport equation
\[ \begin{aligned} &\frac{\partial a}{\partial t}+(v\cdot\nabla)a=f,\\ &a| _{t=0}=a_0(x). \end{aligned} \]

MSC:
35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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