TY  - JOUR
ID  - DallaRiva2012
LB  - DallaRiva2012
AU  - Dalla Riva, Matteo
TI  - Energy integral of the Stokes flow in a singularly perturbed exterior domain
ST  - Energy integral of the Stokes flow in a singularly perturbed exterior domain
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2012
VL  - 32
IS  - 4
SP  - 647
EP  - 659
DO  - http://dx.doi.org/10.7494/OpMath.2012.32.4.647
UR  - http://dx.doi.org/10.7494/OpMath.2012.32.4.647
AB  - 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 \(\epsilon ^{-1}\), \(\log \epsilon\).
KW  - boundary value problem for the Stokes system
KW  - singularly perturbed exterior domain
KW  - real analytic continuation in Banach space
SN  - 1232-9274
ER  -