具有扩散温度的二维Boussinesq方程的Hölder连续弱解
Hölder continuous weak solution of 2d Boussinesq equation with diffusive temperature
摘要：
证明了具有扩散温度的二维Boussinesq方程在给定动能下存在Hölder连续周期弱解.更确切地说，对于任何光滑的$e(t):[0,1]\rightarrow R_+$和$\varepsilon\in (0, \frac{1}{10})$，存在$v\in C^{\frac{1}{10}-\varepsilon}([0,1]\times {\rm T}^2), θ\in C_t^{1,\frac{1}{20}-\frac{\varepsilon}{2}}C_x^{2,\frac{1}{10}-\varepsilon}([0,1]\times {\rm T}^2)$在分布意义下解Boussinesq方程，并且满足e（t）= \int_{{\rm T}^2}| v（t，x）|^2dx，\quad \forall t \in [0，1].
英文摘要：
We show the existence of Hölder continuous periodic weak solutions of the 2d Boussinesq equation with diffusive temperature which satisfy the prescribed kinetic energy. More precisely, for any smooth $e(t):[0,1]\rightarrow R_+$ and $\varepsilon\in (0, \frac{1}{10})$, there exist $v\in C^{\frac{1}{10}-\varepsilon}([0,1]\times {\rm T}^2), θ\in C_t^{1,\frac{1}{20}-\frac{\varepsilon}{2}}C_x^{2,\frac{1}{10}-\varepsilon}([0,1]\times {\rm T}^2)$ which solve boussinesq equation in the sense of distribution and satisfy e(t)=\int_{{\rm T}^2}|v(t,x)|^2dx, \quad \forall t\in [0,1].
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