On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the freeboundary problem of a compressible barotropic viscous fluid flow without surface tensionin the L_p in time and L_q in space framework with 2 < p < ∞ and N < q < ∞ under the assumption that the initial domain is a uniform W^2-1/q_q one in ℝ^N (N ≥ 2).After transforming a unknown time dependent domain to the initial domain by theLagrangian transformation, we solve problem by the Banach contraction mappingprinciple based on the maximal L_p-L_q regularity of the generalized Stokes operatorfor the compressible viscous fluid flowwith free boundary condition. The key issue forthe linear theorem is the existence of R-bounded solution operator in a sector, whichcombined with Weis's operator valued Fourier multiplier theorem implies the generationof analytic semigroup and the maximal L_p-L_q regularity theorem. The nonlinearproblem we studied here was already investigated by several authors (Denisova andSolonnikov, St. Petersburg Math J 14:1-22, 2003; J Math Sci 115:2753-2765, 2003; Secchi, Commun PDE 1:185-204, 1990; Math Method Appl Sci 13:391-404, 1990;Secchi and Valli, J Reine Angew Math 341:1-31, 1983; Solonnikov and Tani, Constantincarathéodory: an international tribute, vols 1, 2, pp 1270-1303,World ScientificPublishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin,1992; Tani, J Math Kyoto Univ 21:839-859, 1981; Zajaczkowski, SIAM JMath Anal25:1-84, 1994) in the L2 framework and Hölder spaces, but our approach is differentfrom them.