Opuscula Math. 32, no. 4 (2012), 647-659
http://dx.doi.org/10.7494/OpMath.2012.32.4.647

Opuscula Mathematica

# Energy integral of the Stokes flow in a singularly perturbed exterior domain

Matteo Dalla Riva

Abstract. We consider a pair of domains $$\Omega ^b$$ and $$\Omega ^s$$ in $$\mathbb{R}^n$$ and we assume that the closure of $$\Omega ^b$$ does not intersect the closure of $$\epsilon \Omega ^s$$ for $$\epsilon \in (0,\epsilon _0)$$. Then for a fixed $$\epsilon \in (0,\epsilon_0)$$ we consider a boundary value problem in $$\mathbb{R}^n \setminus (\Omega ^b \cup \epsilon \Omega ^s)$$ which describes the steady state Stokes flow of an incompressible viscous fluid past a body occupying the domain $$\Omega ^b$$ and past a small impurity occupying the domain $$\epsilon \Omega ^s$$. The unknown of the problem are the velocity field $$u$$ and the pressure field $$p$$, and we impose the value of the velocity field $$u$$ on the boundary both of the body and of the impurity. We assume that the boundary velocity on the impurity displays an arbitrarily strong singularity when $$\epsilon$$ tends to 0. The goal is to understand the behaviour of the strain energy of $$(u, p)$$ for $$\epsilon$$ small and positive. The methods developed aim at representing the limiting behaviour in terms of analytic maps and possibly singular but completely known functions of $$\epsilon$$, such as $$\epsilon ^{-1}$$, $$\log \epsilon$$.

Keywords: boundary value problem for the Stokes system, singularly perturbed exterior domain, real analytic continuation in Banach space.

Mathematics Subject Classification: 76D07, 35J57, 31B10, 45F15.

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