Some free boundary problem for two-phase inhomogeneous incompressible flows

Hirokazu Saito, Yoshihiro Shibata, Xin Zhang

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


In this paper, we establish some local and global solutions for the two-phase incompressible inhomogeneous flows with moving interfaces in the maximal Lp - Lq regularity class. Compared with previous results obtained by Solonnikov [Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), pp. 1065-1087, 1118 (in Russian); translation in Math. USSR-Isz., 31 (1988), pp. 381-405] and by Shibata and Shimizu [Differential Integral Equations, 20 (2007), pp. 241-276], we find the local solutions in the Lp - Lq class in some general uniform Wr2 - 1/r domain in R N by assuming (p, q) ∊]2, ∞[×]N, ∞[or (p, q) ∊]1, 2[×]N, ∞[satisfying 1/p + N/q > 3/2. In particular, the initial data with less regularity 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.

Original languageEnglish
Pages (from-to)3397-3443
Number of pages47
JournalSIAM Journal on Mathematical Analysis
Issue number4
Publication statusPublished - 2020


  • Analytic semigroup
  • Inhomogeneous incompressible navier-stokes equations
  • Maximal lp - lq regularity
  • Two-phase problem

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics


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