Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space?

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Abstract

An important open problem in the theory of the Navier-Stokes equations is the uniqueness of the Leray-Hopf weak solutions with L2 initial data. In this paper we give sufficient conditions for non-uniqueness in terms of spectral properties of a natural linear operator associated to scale-invariant solutions recently constructed in [8]. If the spectral conditions are satisfied, non-uniqueness and ill-posedness can appear for quite benign compactly supported data, just at the borderline of applicability of the classical perturbation theory. The verification of the spectral conditions seems to be approachable by numerical simulations which involve only smooth functions.

Original languageEnglish (US)
Pages (from-to)3734-3766
Number of pages33
JournalJournal of Functional Analysis
Volume268
Issue number12
DOIs
StatePublished - Jun 15 2015

Bibliographical note

Funding Information:
The work of both authors on this project was supported in part by NSF-DMS grants 1101428 and 1159376 .

Keywords

  • Navier Stokes equations
  • Possible non-uniqueness of Leray Hopf weak solutions
  • Scale invariant solutions

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