Maximal L_p-L_q regularity for the two-phase Stokes equations; Model problems

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 R boundedness of solution operators defined in a sector Σ_ε,γ0={λεC\{0}||argλ| ≤ π-ε,|λ|≥γ0} with 0<ε<π/2 and γ0≥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(Ṙⁿ,Σ_ε,γ0) and E_p,q,γ0(Ṙn×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.