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Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. (English) Zbl 0819.35109
Summary: A new approach to resolvent estimates of the Stokes equations in \(L^ q\)-spaces, \(1<q<\infty\), for several classes of unbounded domains and for bounded domains is developed. Compared with the literature this approach is more elementary, includes a new class of domains with noncompact boundary and admits a nonzero divergence \(g= \text{div } u\) which is prescribed arbitrarily in some function class.
First the authors prove resolvent estimates for the whole space \(\mathbb{R}^ n\) and the half space \(\mathbb{R}^ n_ +\) via multiplier techniques. By perturbation, duality and localization arguments, they get analogous results for perturbed half spaces, cones, exterior and bounded domains the boundary of which is assumed to be of class \(C^{1,1}\). The results imply that the Stokes operator generates a (bounded) analytic semigroup of these classes of domains. Another application yields new a priori estimates for the well known problem \(\text{div } u=g\) with zero boundary conditions of \(u\).

MSC:
35Q30 Navier-Stokes equations
47D03 Groups and semigroups of linear operators
76D07 Stokes and related (Oseen, etc.) flows
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