Some free boundary problem for two phase inhomogeneous incompressible flow

TL;DR

This paper establishes local and global solutions for two-phase inhomogeneous incompressible flows with moving interfaces, extending previous results by allowing less regular initial data and constructing long-term solutions under specific conditions.

Contribution

It introduces new existence results in $L_p-L_q$ spaces for less regular initial data and constructs long-term solutions for bounded droplets with piecewise constant density and viscosity.

Findings

Established local solutions in $L_p-L_q$ class in general domains.

Allowed less regular initial data by considering $p<2$.

Constructed long-time solutions for bounded droplets with decay properties.

Abstract

In this paper, we establish some local and global solutions for the two phase incompressible inhomogeneous flows with moving interfaces in $L_p-L_q$ maximal regularity class. Compared with previous results obtained by V.A.Solonnikov and by Y.Shibata \& S.Shimizu, we find the local solutions in $L_p-L_q$ class in some general uniform $W^{2-1\slash r}_r$ domain in $\BBR^N$ by assuming $(p, q)\in ]2,\infty[\times ]N,\infty[$ or $(p, q)\in ]1,2[ \times ]N,\infty[$ satisfying $1 \slash p + N \slash q>3\slash 2.$ In particular, less regular initial data are allowed by assuming $p<2.$ In addition, if the density and the viscosity coefficient are piecewise constant, we can construct the long time solution from the small initial states in the case of the bounded droplet. This is due to some decay property for the corresponding linearized problem.