Global existence and uniqueness of strong solutions to the 2D nonhomogeneous primitive equations with density-dependent viscosity

Abstract

This paper is concerned with an initial-boundary value problem of the two-dimensional inhomogeneous primitive equations with density-dependent viscosity. The global well-posedness of strong solutions is established, provided the initial horizontal velocity is suitably small, that is, $\|\nabla u_{0}\|_{L^{2}}\leq \eta_{0}$ for suitably small $\eta_{0}>0$. The initial data may contain vacuum. The proof is based on the local well-posedness and the blow-up criterion proved in \cite{0}, which states that if $T^{*}$ is the maximal existence time of the local strong solutions $(\rho,u,w,P)$ and $T^{*}<\infty$, then \begin{equation*} \sup_{0\leq t<T^{*}}(\left\|\nabla \rho(t)\right\|_{L^{\infty}}+\left\|\nabla^{2}\rho(t)\right\|_{L^{2}}+\left\|\nabla u(t)\right\|_{L^{2}})=\infty. \end{equation*} To complete the proof, it is required to make an estimate on a key term $\|\nabla u_{t}\|_{L_{t}^{1}L_{\Omega}^{2}}$. We prove that it is bounded and could be as small as desired under certain smallness conditions, by making use of the regularity result of hydrostatic Stokes equations and some careful time weighted estimates.