Besov空间中Boussinesq方程的弱解的正则性
On the regularity of weak solutions of the Boussinesq equations in Besov spaces Dedicated to Enrique Zuazua on the occasion of his sixtieth birthday
摘要：
本文讨论的主要问题是Z. Zhang在齐次Besov空间$\dot{B}_{\infty ,\infty }^{-1}(\mathbb{R}% ^{3})$的上下文中证明，如果下面的Boussinesq方程（\ref % {eq1.1}）的解（从$H^{2}$中的初始数据开始）使得$% (\nabla u,\nabla θ)\in L^{2}\left( 0,T;\dot{B}_{\infty ,\infty }^{-1}(% \mathbb{R}^{3})\right)$，则该解在$T$之后永远保持光滑。在这篇文章中，我们证明了相同的结果弱解，只是假设的条件上的速度$u$，而不是温度$θ$。
英文摘要：
The main issue addressed in this paper concerns an extension of a result by Z. Zhang who proved, in the context of the homogeneous Besov space $\dot{B}_{\infty ,\infty }^{-1}(\mathbb{R}% ^{3})$, that, if the solution of the Boussinesq equation (\ref% {eq1.1}) below (starting with an initial data in $H^{2}$) is such that $% (\nabla u,\nabla θ)\in L^{2}\left( 0,T;\dot{B}_{\infty ,\infty }^{-1}(% \mathbb{R}^{3})\right)$, then the solution remains smooth forever after $T$. In this contribution, we prove the same result for weak solutions just by assuming the condition on the velocity $u$ and not on the temperature $θ$.
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