| Abstract | The aim of this paper is to show the existence of R-bounded solution operator families for two-phase Stokes resolvent equations in Omega = Omega(+)boolean OR Omega(-), where Omega(+/-) are uniform W-r(2-1/r) domains of N-dimensional Euclidean space R-N (N >= 2, N < r < infinity). More precisely, given a uniform W-r(2-1/r) domain Omega with two boundaries Gamma(+/-) satisfying Gamma(+)boolean AND Gamma(-) = phi, we suppose that some hypersurface Gamma divides Omega into two sub-domains, that is, there exist domains Omega(+/-) subset of Omega such that Omega(+) boolean AND Omega(-)= phi and Omega \ Gamma = Omega(+) boolean OR Omega(-), where Gamma boolean AND Gamma(+) = phi, Gamma boolean AND Gamma(-) = phi, and the boundaries of Omega(+/-) consist of two parts Gamma and Gamma(+/-) respectively. The domains Omega(+/-) are filled with viscous, incompressible, and immiscible fluids with density rho(+/-) and viscosity mu(+/-), respectively. Here, rho(+/-) are positive constants, while mu(+/-) = mu(+/-)(x) are functions of x is an element of R-N. On the boundaries Gamma, Gamma(+), and Gamma(-), we consider an interface condition, a free boundary condition, and the Dirichlet boundary condition, respectively. We also show, by using the R,-bounded solution operator families, some maximal L-p-L-q regularity as well as generation of analytic semigroup for a time-dependent problem associated with the two-phase Stokes resolvent equations. This kind of problems arises in the mathematical study of the motion of two viscous, incompressible, and immiscible fluids with free surfaces. The essential assumption of this paper is the unique solvability of a weak elliptic transmission problem for f is an element of L-q(Omega)(N), that is, it is assumed that the unique existence of solutions theta is an element of W-q(1)(Omega) to the variational problem: (rho(-1) del theta, del phi)(Omega) = (f, del phi)(Omega) for any phi is an element of W-q(1)(Omega) with 1 < q < infinity and q' = q/(q - 1), where rho is defined by rho = rho(+) (x is an element of Omega(+)), rho = rho(-) (x is an element of Omega(-)) and W-q(1)(Omega) is a suitable Banach space endowed with norm parallel to . parallel to(Wq1(Omega)) : = parallel to del . parallel to(Lq(Omega)). Our assumption covers e.g. the following domains as Omega: R-N, R-+/-(N), perturbed R-+/-(N), layers, perturbed layers, and bounded domains, where R-+(N) and R--(N) are the open upper and lower half spaces, respectively. |
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