The Navier–Stokes Equations in Nonendpoint Borderline Lorentz Spaces

Abstract

It is shown both locally and globally that \({L_t^{\infty}(L_x^{3,q})}\) solutions to the three-dimensional Navier–Stokes equations are regular provided \({q\neq\infty}\). Here \({L_x^{3,q}}\), \({0 < q \leq\infty}\), is an increasing scale of Lorentz spaces containing \({L^3_x}\). Thus the result provides an improvement of a result by Escauriaza et al. (Uspekhi Mat Nauk 58:3–44, 2003; translation in Russ Math Surv 58, 211–250, 2003), which treated the case q = 3. A new local energy bound and a new \({\epsilon}\)-regularity criterion are combined with the backward uniqueness theory of parabolic equations to obtain the result. A weak-strong uniqueness of Leray–Hopf weak solutions in \({L_t^{\infty}(L_x^{3,q})}\), \({q\neq\infty}\), is also obtained as a consequence.

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Correspondence to Nguyen Cong Phuc.

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Communicated by R. Shvydkoy

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Phuc, N.C. The Navier–Stokes Equations in Nonendpoint Borderline Lorentz Spaces. J. Math. Fluid Mech. 17, 741–760 (2015). https://doi.org/10.1007/s00021-015-0229-2

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Mathematics Subject Classification

  • 35Q30

Keywords

  • Navier–Stokes equations
  • partial regularity
  • backward uniqueness
  • Lorentz space
  • Ladyzhenskaya–Prodi–Serrin type condition