Smoothing effect and well-posedness for 2D Boussinesq equations in critical Sobolev space

Abstract

<p style='text-indent:20px;'>In this paper, we investigate the fractional dissipation 2D Boussinesq equations with initial data in the critical space <inline-formula><tex-math id="M1">\begin{document}$ u_0\in H^{2-2\alpha}(\mathbb{R}^2) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \theta_0\in H^{2-2\beta}(\mathbb{R}^2) $\end{document}</tex-math></inline-formula>. The local well-posedness for the equations is firstly established by using some <i>a priori</i> estimates for the solution in <inline-formula><tex-math id="M3">\begin{document}$ L^{p}(0, T;{H}^{2-\frac{p-1}{p} 2\alpha}(\mathbb{R}^2))\times L^{p}(0, T;{H}^{2-\frac{p-1}{p} 2\beta}(\mathbb{R}^2)) $\end{document}</tex-math></inline-formula> with some suitable <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula>. And then the generalized blow-up criterion and smoothing effect are obtained in turn, which improves some of the previous results for (critical, subcritcial or supcritical) Boussnesq equations. The results of the present paper are based on the Littlewood-Paley theory and the nonlinear lower bounds estimates for the fractional Laplacian, and can be treated as a generalization of results for 2D quasi-geostrophic equation.</p>