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Very weak solutions and the Fujita-Kato approach to the Navier-Stokes system in general unbounded domains

  • Reinhard FarwigEmail author
  • Paul Felix Riechwald
Article

Abstract

We consider the instationary Navier-Stokes system in general unbounded domains \({\Omega \subset \mathbb{R}^{n}}\), \({n\geq 3}\), with smooth boundary and construct by the Fujita-Kato method mild solutions \({u\in L^{\infty}(0,T; \tilde{L}^{n}(\Omega))}\) with initial value \({u_0\in\tilde{L}^{n}(\Omega)}\). Here the classical \({L^n(\Omega)}\)–space is replaced by \({\tilde{L}^n(\Omega)}\) where for q > 2 the space \({\tilde{L}^q}\) is defined by \({L^q\cap L^2}\). Moreover, for suitable initial values we identify mild solutions in \({L^\infty(0,T;\tilde{L}^n(\Omega))}\) with very weak solutions in Serrin’s class \({L^r (0,T;\tilde{L}^q(\Omega))}\) where \({\frac{2}{r} + \frac{n}{q} =1}\), \({2< r < \infty}\).

Keywords

Navier-Stokes equations Fujita-Kato method Mild solutions Very weak solutions General unbounded domains Spaces \({\tilde{L}^{q}(\Omega)}\) 

Mathematics Subject Classification

35B65 76D05 76D03 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.International Research Training Group (IRTG 1529) Darmstadt-TokyoDarmstadtGermany

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