Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in ℜⁿ

Authors: Farwig, R.¹; Sohr, H.²

Source: Czechoslovak Mathematical Journal, Volume 59, Number 1, March 2009 , pp. 61-79(19)

Publisher: Springer

Buy & download fulltext article:

Price: $47.00 plus tax (Refund Policy)

Abstract:

For a bounded domain Ω ⊂ ℜⁿ, n 3, we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system − Δu + u · ∇u + ∇p = f, div u = k, u _|aΩ = g with u ∈ L ^q, q n, and very general data classes for f, k, g such that u may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of papers by Frehse & Růžička, see e.g. Existence of regular solutions to the stationary Navier-Stokes equations, Math. Ann. 302 (1995), 669-717, where the existence of a weak solution which is locally regular is proved.

Keywords: stationary Stokes and Navier-Stokes system; very weak solutions; existence and uniqueness in higher dimensions; regularity classes in higher dimensions; 76D05; 35J55; 35J65; 35Q30; 76D07

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10587-009-0005-7

Affiliations: 1: Fachbereich Mathematik, Technische Universität Darmstadt, D-642 83, Darmstadt, Germany, Email: farwig@mathematik.tu-darmstadt.de 2: Fakultät für Elektrotechnik, Informatik und Mathematik, Universität Paderborn, D-330 98, Paderborn, Germany, Email: hsohr@math.uni-paderborn.de

Publication date: 2009-03-01