Abstract
In this paper, we consider some two phase problems of compressible and incompressible viscous fluids’ flow without surface tension under the assumption that the initial domain is a uniform Wq2−1/q domain in RN (N≥2). We prove the local in the time unique existence theorem for our problem in the Lp in time and Lq in space framework with 2<p<∞ and N<q<∞ under our assumption. In our proof, we first transform an unknown time-dependent domain into the initial domain by using the Lagrangian transformation. Secondly, we solve the problem by the contraction mapping theorem with the maximal Lp-Lq regularity of the generalized Stokes operator for the compressible and incompressible viscous fluids’ flow with the free boundary condition. The key step of our proof is to prove the existence of an R-bounded solution operator to resolve the corresponding linearized problem. The Weis operator-valued Fourier multiplier theorem with R-boundedness implies the generation of a continuous analytic semigroup and the maximal Lp-Lq regularity theorem.
Highlights
It is an important mathematical problem to consider the unsteady motion of a bubble in an incompressible viscous fluid or that of a drop in a compressible viscous one
The problem is, in general, formulated mathematically by the Navier–Stokes equations in a timedependent domain separated by an interface, where one part of the domain is occupied by a compressible viscous fluid and another part by an incompressible viscous fluid
Let Γ ⊂ Ω be a given surface that bounds the region Ω+ occupied by a compressible barotropic viscous fluid and the region Ω− occupied by an incompressible viscous one
Summary
It is an important mathematical problem to consider the unsteady motion of a bubble in an incompressible viscous fluid or that of a drop in a compressible viscous one. To state our theorem on the local in time unique existence of solutions to Equations (1)–(3) and (5), we introduce some functional spaces and the definition of the uniform Wr2−1/r domain. There exists a T > 0 depending on R such that Equation (1) subject to the interface condition (2), boundary condition (3), kinematic condition (4), and initial condition (5) admits a unique solution (ρ+ , v± ) with:. To state our main result for linear Equation (22), we introduce more symbols and functional spaces used throughout this paper. The R-bounded solution operators yield the generation of the continuous analytic semigroup associated with Equation (22), which, combined with some real interpolation technique, yields the L p -Lq maximal regularity for the initial problem for Equation (22) Combining these two results gives Theorem 3.
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