Some free boundary problem for two phase inhomogeneous incompressible flow

2018/11/06 by Hirokazu Saito, Yoshihiro Shibata, Saito, Hirokazu +3 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1811.02179

openalex publication_date 2018/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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)∈ ]2,∞[× ]N,∞[ or (p, q)∈ ]1,2[ × ]N,∞[ 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.

Cited by

• On 2D Navier-Stokes free boundary: nonnegative density and small viscosity contrast

Related

• Global solvability of compressible-incompressible two-phase flows with phase transitions in bounded domains
• Strong solutions to compressible-incompressible two-phase flows with phase transitions
• Local well-posedness of compressible-incompressible two-phase flows with phase transitions
• On the \mathcal R-boundedness for the two phase problem with phase transition: compressible-incompressible model problem
• Global-in-time Discontinuous Solutions for the Two-Phase Model of Compressible Fluids with Density-Dependent Viscosity