Abstract
Consider a free boundary problem of compressible-incompressible two-phase flows with surface tension and phase transition in bounded domains Ωt+,Ωt−⊂RN, N≥2, where the domains are separated by a sharp compact interface Γt⊂RN−1. We prove a global in time unique existence theorem for such free boundary problem under the assumption that the initial data are sufficiently small and the initial domain of the incompressible fluid is close to a ball. In particular, we obtain the solution in the maximal Lp−Lq-regularity class with 2<p<∞ and N<q<∞ and exponential stability of the corresponding analytic semigroup on the infinite time interval.
Highlights
The purpose of this paper is to prove the global solvability of the free boundary problem of compressible–incompressible two-phase flows with phase transitions in bounded domains
Two fluids are separated by a free boundary Γt and a surface tension and phase transitions are taken into account
Our problem is formulated as follows: Let Ω be a bounded domain in N-dimensional Euclidean space R N, N ≥ 2, surrounded by a smooth boundary Γ+
Summary
The purpose of this paper is to prove the global solvability of the free boundary problem of compressible–incompressible two-phase flows with phase transitions in bounded domains. We further assume the following properties: The pressure field p+ (ρ+ ) is a C2 -function defined on ρ∗+ /3 ≤ ρ+ ≤ 3ρ∗+ such that 0 < p0+ (ρ+ ) ≤ π ∗ with some positive constant π ∗ for any ρ∗+ /3 ≤ ρ+ ≤ 3ρ∗+. The conditions (9) are imposed to avoid multiple roots of the characteristic equation arising in the model problems in the half space and the whole space with flat interface In those cases, applying the partial Fourier transform to the generalized resolvent problem yields the ODEs with respect to x N , and the solution formula is obtained by the inverse partial Fourier transform, see Section 2 of [14] and Section 4 of [8].
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