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
Basson, A.: Solutions spatialement homogènes adaptées des équations de Navier–Stokes. Université d’Évry, Thèse 2006
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)
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
Grafakos, L.: Classical Harmonic Analysis, 2nd edn. Springer, Berlin 2008
Grafakos, L.: Modern Harmonic Analysis, 2nd edn. Springer, Berlin 2009
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
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
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
Lemarié-Rieusset, P.G.: Recent Developments in the Navier–Stokes Problem. CRC Press, Boca Raton 2002
Lemarié-Rieusset, P.G.: The Navier–Stokes Problem in the 21st Century. Chapman & Hall/CRC, New York 2016
Leray, J.: Essai sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math. 63, 193–248, 1934
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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