Affiliations: TU Darmstadt, Darmstadt, Germany. E-mail: [email protected] | Department of Mathematics and Research Institute of Science and Engineering, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan. E-mail: [email protected]
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 Lp–Lq 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 Lp in time and Lq 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.
