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Weak Solutions for Navier–Stokes Equations with Initial Data in Weighted \(L^2\) Spaces

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Abstract

We show the existence of global weak solutions to the three dimensional Navier–Stokes equations with initial velocity in the weighted spaces \(L^2_{w_\gamma }\), where \(w_\gamma (x)=(1+\vert x\vert )^{-\gamma }\) and \(0<\gamma \leqq 2\), using new energy controls. As an application we give a new proof of the existence of global weak discretely self-similar solutions to the three dimensional Navier–Stokes equations for discretely self-similar initial velocities which are locally square integrable.

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References

  1. Basson, A.: Solutions spatialement homogènes adaptées des équations de Navier–Stokes. Université d’Évry, Thèse 2006

  2. Bradshaw, Z., Tsai, T.P.: Discretely self-similar solutions to the Navier–Stokes equations with data in \(L^2_{\rm loc}\) (to appear in Analysis and PDE)

  3. Chae, D., Wolf, J.: Existence of discretely self-similar solutions to the Navier–Stokes equations for initial value in \(L^2_{\rm loc}({\mathbb{R}}^3)\). Ann. Inst. H. Poincaré Anal. Non Linéaire35, 1019–1039, 2018

    Article  ADS  MathSciNet  Google Scholar 

  4. Grafakos, L.: Classical Harmonic Analysis, 2nd edn. Springer, Berlin 2008

    MATH  Google Scholar 

  5. Grafakos, L.: Modern Harmonic Analysis, 2nd edn. Springer, Berlin 2009

    MATH  Google Scholar 

  6. Jia, H., Šverák, V.: Local-in-space estimates near initial time for weak solutions of the Navier–Stokes equations and forward self-similar solutions. Invent. Math. 196, 233–265, 2014

    Article  ADS  MathSciNet  Google Scholar 

  7. Kikuchi, N., Seregin, G.: Weak solutions to the Cauchy problem for the Navier–Stokes equations satisfying the local energy inequality, in Nonlinear equations and spectral theory. Amer. Math. Soc. Transl. Ser. Vol. 2, No. 220 (Eds. Birman M.S. and Uraltseva N.N.), 141–164, 2007

  8. Lemarié-Rieusset, P.G.: Solutions faibles d’énergie infinie pour les équations de Navier–Stokes dans \({\mathbb{R}}^{3}\). C. R. Acad. Sci. Paris, Serie I. 328, 1133–1138, 1999

  9. Lemarié-Rieusset, P.G.: Recent Developments in the Navier–Stokes Problem. CRC Press, Boca Raton 2002

    Book  Google Scholar 

  10. Lemarié-Rieusset, P.G.: The Navier–Stokes Problem in the 21st Century. Chapman & Hall/CRC, New York 2016

    Book  Google Scholar 

  11. Leray, J.: Essai sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math. 63, 193–248, 1934

    Article  MathSciNet  Google Scholar 

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Correspondence to Pedro Gabriel Fernández-Dalgo.

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Communicated by V. Šverák

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Fernández-Dalgo, P.G., Lemarié-Rieusset, P.G. Weak Solutions for Navier–Stokes Equations with Initial Data in Weighted \(L^2\) Spaces. Arch Rational Mech Anal 237, 347–382 (2020). https://doi.org/10.1007/s00205-020-01510-w

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  • DOI: https://doi.org/10.1007/s00205-020-01510-w

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