标题： 关于具有热扩散的三维Boussinesq方程的 Hölder 连续弱解 显示英文标题

标题： Hölder continuous weak solutions of the 3D Boussinesq equation with thermal diffusion

摘要： 本文中，我们展示了具有热扩散的三维Boussinesq方程存在Hölder连续周期弱解，这些解逼近Onsager临界空间正则性，并满足规定的动能。 更确切地说，对于任意光滑的$e(t):[0,T]\rightarrow \mathbb{R}_+$和$\beta\in (0, \frac{1}{3})$，存在$v\in C^{\beta}([0,T]\times {\mathbb{T} }^3)$和$ \theta\in C_t^{1,\frac{\beta}{2}}C_x^{2,\beta}([0,T]\times {\mathbb{T} }^3)$在分布意义下求解(\ref{e:boussinesq equation})并满足\begin{align} e(t)=\int_{{{\mathbb{T} }^3}}|v(t,x)|^2dx, \quad \forall t\in [0,T].\nonumber \end{align}。 显示英文摘要

摘要： In this paper, we show the existence of H\"{o}lder continuous periodic weak solutions of the 3D Boussinesq equation with thermal diffusion, which apprroximate the Onsager's critical spatial regularity and satisfy the prescribed kinetic energy. More precisely, for any smooth $e(t):[0,T]\rightarrow \mathbb{R}_+$ and $\beta\in (0, \frac{1}{3})$, there exist $v\in C^{\beta}([0,T]\times {\mathbb{T} }^3)$ and $ \theta\in C_t^{1,\frac{\beta}{2}}C_x^{2,\beta}([0,T]\times {\mathbb{T} }^3)$ which solve (\ref{e:boussinesq equation}) in the sense of distribution and satisfy \begin{align} e(t)=\int_{{{\mathbb{T} }^3}}|v(t,x)|^2dx, \quad \forall t\in [0,T].\nonumber \end{align}