On some free boundary problem of the Navier-Stokes equations in the maximal $L_p$-$L_q$ regularity class

Abstract

This paper is concerned with the free boundary problem for the Navier Stokes equations without surface tension in the $L_p$ in time and $L_q$ in space setting with $2<p<\infty$ and $N<q<\infty$. A local in time existence theorem is proved in a uniform $W_q^{2-1/q}$ domain in the $N$-dimensional Euclidean space ${\mathbb{R}}^N$ ($N\ge 2$) under the assumption that weak Dirichlet-Neumann problem is uniquely solvable. Moreover, a global in time existence theorem is proved for small initial data under the assumption that $\Omega$ is bounded additionally. This was already proved by Solonnikov \cite{Sol1} by using the continuation argument of local in time solutions which are exponentially stable in the energy level under the assumption that the initial data is orthogonal to the rigid motion. We also use the continuation argument and the same orthogonality for the initial data. But, our argument about the continuation of local in time solutions is based on some decay theorem for the linearized problem, which is a different point than \cite{Sol1}.