Maximal Lp-Lq regularity for the two-phase Stokes equations; Model problems

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:251

Article

Shibata, Yoshihiro²  Shimizu, Senjo¹ 

[1] Shizuoka Univ, Fac Sci, Dept Math, Shizuoka 4228529, Japan

[2] Waseda Univ, Dept Math, Res Inst Sci & Engn, Shinjuku Ku, Tokyo 1698555, Japan

关键词: Stokes equation;    Two-phase problem;    Maximal regularity;    Resolvent estimate;    Surface tension;    Gravity;   

DOI  :  10.1016/j.jde.2011.04.005

来源: Elsevier

【 摘 要 】

In this paper we prove the generalized resolvent estimate and maximal L-p-L-q regularity of the Stokes equation with and without surface tension and gravity in the whole space with flat interface. We prove II boundedness of solution operators defined in a sector Sigma(epsilon,gamma 0) = {lambda is an element of C\ {0} vertical bar vertical bar arg lambda vertical bar <= pi - epsilon vertical bar lambda vertical bar >= gamma(0)] with 0 < epsilon < pi/2 and gamma(0) >= 0, which combined with the Fourier multiplier theorem of S.G. Mihlin and the operator valued Fourier multiplier theorem of L Weis yields the required generalized resolvent estimate and maximal L-p-L-q regularity at the same time. One of the character of the paper is to introduce special function spaces E-q((R) over dot(n),Sigma(epsilon,gamma 0) and E-p,E-q,E-gamma 0 ((R) over dot(n) x R) (cf. (1.7) and (1.8)), which is necessary to treat the situation that the normal component of velocity fields jumps across the interface. Such spaces never appear in the study of the Stokes equations with other boundary conditions like non-slip condition, Navier slip condition, Robin condition or pure Neumann condition appearing in the study of one phase problem (cf. Desch et al., 2001 [12], Farwig and Sohr, 1994 [13], Saal, 2003 [22], Shibata and Shimada, 2007 [23], Shibata and Shimizu 2008 [25], 2009 [26], in press [27]), because the normal component of the velocity fields vanishes at the boundary which is physical requirement that the flow does not go out and come in through the rigid boundary. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

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