TY - JOUR

T1 - Some free boundary problem for two-phase inhomogeneous incompressible flows

AU - Saito, Hirokazu

AU - Shibata, Yoshihiro

AU - Zhang, Xin

N1 - Funding Information:
\ast Received by the editors November 6, 2018; accepted for publication (in revised form) April 15, 2020; published electronically July 27, 2020. https://doi.org/10.1137/18M1225239 Funding: The first author was supported by JSPS KAKENHI grant JP17K14224. The second author was supported by JSPS KAKENHI grant JP17H0109 and by the Top Global University Project. \dagger Department of Mathematics, The University of Electro-Communications, Tokyo, Japan (hsaito@uec.ac.j). \ddagger Department of Mathematics, School of Fundamental Science and Engineering, Waseda University, Tokyo, Japan, and Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15260 (yshibata@waseda.jp). \S Waseda Research Institute for Science and Engineering, Faculty of Science and Engineering, Waseda University, Tokyo, Japan (xin.zhang1988@gmail.co).

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - Analytic semigroup

KW - Inhomogeneous incompressible navier-stokes equations

KW - Maximal lp - lq regularity

KW - Two-phase problem

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U2 - 10.1137/18M1225239

DO - 10.1137/18M1225239

M3 - Article

AN - SCOPUS:85091830246

VL - 52

SP - 3397

EP - 3443

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 4

ER -