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Existence of Suitable Weak Solutions to the Navier–Stokes Equations for Intermittent Data

  • Zachary BradshawEmail author
  • Igor Kukavica
Article
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Abstract

Local in time weak solutions to the 3D Navier–Stokes are constructed for a class of initial data in \(L^2_\mathrm {loc}\). In contrast to other constructions (e.g. Lemarié-Rieusset in Recent developments in the Navier–Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, vol 431. Chapman & Hall/CRC, Boca Raton, 2002; Kikuchi and Seregin in Weak solutions to the Cauchy problem for the Navier–Stokes equations satisfying the local energy inequality. Nonlinear equations and spectral theory, American Mathematical Society translations: series 2, vol 220. American Mathematical Society, Providence, pp 141–164, 2007; Kwon and Tsai in Global Navier–Stokes flows for non-decaying initial data with slowly decaying oscillation. arXiv:1811.03249 ), the initial data is not required to be uniformly locally square integrable and, in particular, can exhibit growth in a local \(L^2\) sense. This class of initial data includes vector fields in the critical Morrey space and discretely self-similar vector fields in \(L^2_\mathrm {loc}\).

Mathematics Subject Classification

35Q30 76D05 

Notes

Acknowledgements

ZB was supported in part by the Simons Foundation, while IK was supported in part by the NSF Grants DMS-1615239 and DMS-1907992.

Compliance with ethical standards

Conflict of interest

The authors have no conflict of interest to report.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ArkansasFayettevilleUSA
  2. 2.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

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