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

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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.

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, D-642 83, Darmstadt, Germany

    R. Farwig

  2. Fakultät für Elektrotechnik, Informatik und Mathematik, Universität Paderborn, D-330 98, Paderborn, Germany

    H. Sohr

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  1. R. Farwig
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  2. H. Sohr
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Farwig, R., Sohr, H. Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in ℝn . Czech Math J 59, 61–79 (2009). https://doi.org/10.1007/s10587-009-0005-7

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