Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum

Abstract

This paper concerns the Cauchy problem of the two-dimensional density-dependent Boussinesq equations on the whole space \begin{document}$ \mathbb{R}^{2} $\end{document} with zero density at infinity. We prove that there exists a unique global strong solution provided the initial density and the initial temperature decay not too slow at infinity. In particular, the initial data can be arbitrarily large and the initial density may contain vacuum states and even have compact support. Moreover, there is no need to require any Cho-Choe-Kim type compatibility conditions. Our proof relies on the delicate weighted estimates and a lemma due to Coifman-Lions-Meyer-Semmes [J. Math. Pures Appl., 72 (1993), pp. 247-286].