On the ℛ-boundedness of the solution operators in the study of the compressible viscous fluid flow with free boundary conditions

Abstract

In this paper, we consider a generalized resolvent problem for the linearization system of the Navier–Stokes equations describing some free boundary problem of a compressible barotropic viscous fluid flow without taking the surface tension into account. We prove the existence of the ℛ-bounded solution operators, which drives not only the generation of analytic semigroup but also the maximal L_p–L_q regularity by means of Weis' operator valued Fourier multiplier theorem for the corresponding time dependent problem that enable us to prove a local in time existence theorem of the free boundary problem for a compressible barotropic viscous fluid flow in the L_p in time and L_q space setting (cf. Annali dell Universita di Ferrara 60 (2014), 55–89). The results in this paper were given in the PhD thesis [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] by the first author under supervision of the second author. Here we present a slightly different method of deriving a concrete form of solutions to the model problem. In this paper, one of the essential points is to show the invertibility of a 2×2 Lopatinski matrix function. The corresponding system in [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] is a 3×3 matrix, so that the method presented here is slightly simpler.