On the global well-posedness of some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a global in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the L-p in time and L-q in space framework with 2 < p < infinity and N < q < infinity under the assumption that the initial domain is bounded and initial data are small enough and orthogonal to rigid motions. Such global well-posedness was proved by Zajaczkowski in 1993 in the L-2 framework, and our result is an extension of his result to the maximal L-p-L-q regularity setting. We use the maximal L-p-L-q regularity theorem for the linearlized equations and the exponential stability of the corresponding analytic semigroup, which is a completely different approach than Zajaczkowski (1993).