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On the Stokes and Navier‐Stokes System for Domains with Noncompact Boundary in L q ‐spaces
Author(s) -
Farwig Reinhard,
Sohr Hermann
Publication year - 1994
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19941700106
Subject(s) - mathematics , disjoint sets , mathematical analysis , nonlinear system , multiplier (economics) , mathematical proof , space (punctuation) , perturbation (astronomy) , navier–stokes equations , stokes flow , boundary (topology) , pure mathematics , geometry , compressibility , physics , quantum mechanics , linguistics , philosophy , economics , macroeconomics , flow (mathematics) , thermodynamics
We consider the L q ‐theory of weak solutions of the Stokes and Navier‐Stokes equations in two classes of unbounded domains with noncompact boundary, namely in perturbed half spaces which are obtained by a perturbation of the half space IR n , and in aperture domains consisting of two disjoint half spaces separated by a wall but connected by a hole (aperture) through this wall. The proofs rest on the cut‐off procedure and a new multiplier approach to the half space problem. In an aperture domain we additionally prescribe either the flux through the wall or the pressure drop at infinity to single out a unique solution. The nonlinear problem is solved for sufficiently small data and requires q = n/2, n ≥ 3, to estimate the nonlinearity.