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On the \(\mathcal{R}\)-boundedness of solution operator families for two-phase Stokes resolvent equations. (English) Zbl 1424.35277

The authors investigate the existence of so-called \(\mathcal{R}\)-bounded solution operator families for two-phase Stokes resolvent equations in \(\dot{\Omega}={\Omega}_{+}\cup{\Omega}_{-}\), where \({\Omega}_{\pm}\) are uniform \(W_{r}^{2-1/r}\) domains of \(N\)-dimensional Euclidean space \(\mathbb{R}^N\), \((N\geq 2,\ N<r<\infty)\). The domains \({\Omega}_{\pm}\) are filled with viscous, incompressible, and immiscible fluids with different densities and viscosities. On the boundaries the authors use a free boundary condition combined with Dirichlet conditions on the known parts. The authors show 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.
The paper is very comprehensive. The proofs are given in detail and the bibliography which presents a good survey on the topic contains 31 items. The topic is mainly the study of the common motion of two viscous incompressible immiscible fluids. This kind of problems is very close to the problems to be handled in the present paper. Most of the papers in the bibliography were published after the year 2000.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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