Multipliers, paramultipliers, and weak-strong uniqueness for the Navier-Stokes equations

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Abstract

In this article, we describe spaces P such that: if u is a weak (in the sense of Leray [J. Leray, Sur le mouvement d'un fluide visqueux remplissant l'espace, Acta Math. 63 (1934) 193-248]) solution of the Navier-Stokes system for some initial data u0, and if u belongs to P, then u is unique in the class of weak solutions. We say then that weak-strong uniqueness holds. It turns out that the proof of such results relies on the boundedness of a trilinear functional F : L2 / α over(H, ̇)α × L2 / β over(H, ̇)β × P → R, where α, β belong to [ 0, 1 ]. In order to find optimal conditions for the boundedness of F, we are led to describing spaces of multipliers and of paramultipliers (that is, functions which map, by classical pointwise product or by paraproduct, a given Sobolev spaces in another given Sobolev space). The study of these spaces enables us to give conditions for weak-strong uniqueness which generalise all previously known results, from the famous Serrin criterion [J. Serrin, The initial value problem for the Navier-Stokes equations, in: R.E. Langer (Ed.), Nonlinear Problems, Univ. of Wisconsin Press, 1963, pp. 69-98], to the recent conditions formulated by Lemarié-Rieusset [P.-G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman and Hall, 2003].

Original languageEnglish (US)
Pages (from-to)373-428
Number of pages56
JournalJournal of Differential Equations
Volume226
Issue number2
DOIs
StatePublished - Jul 15 2006

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Keywords

  • Leray solutions
  • Multipliers
  • Navier-Stokes equations
  • Paraproduct
  • Sobolev spaces
  • Weak-strong uniqueness

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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