$L_p$-estimates of solution of the free boundary problem for viscous compressible and incompressible fluids in the linear approximation

The paper contains $L_p$-estimates and a theorem on the local in time solvability of the problem arising as a result of linearization of the free boundary problem for two viscous fluids, compressible and incompressible, contained in a bounded vessel, separated by a free interface, and subject to mass and capillary forces. This result is known for the case of $p=2$; it serves as an analytical basis for the study of the complete nonlinear problem. The proof is based on the “maximal regularity” estimate of the solution obtained with the help of the $L_p$ Fourier multiplier theorem due to P. I. Lizorkin.