Self-improving bounds for the Navier-Stokes equations

Jean-Yves Chemin; Fabrice Planchon

Bulletin de la Société Mathématique de France (2012)

  • Volume: 140, Issue: 4, page 583-597
  • ISSN: 0037-9484

Abstract

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We consider regular solutions to the Navier-Stokes equation and provide an extension to the Escauriaza-Seregin-Sverak blow-up criterion in the negative regularity Besov scale, with regularity arbitrarly close to - 1 . Our results rely on turning a priori bounds for the solution in negative Besov spaces into bounds in the positive regularity scale.

How to cite

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Chemin, Jean-Yves, and Planchon, Fabrice. "Self-improving bounds for the Navier-Stokes equations." Bulletin de la Société Mathématique de France 140.4 (2012): 583-597. <http://eudml.org/doc/272604>.

@article{Chemin2012,
abstract = {We consider regular solutions to the Navier-Stokes equation and provide an extension to the Escauriaza-Seregin-Sverak blow-up criterion in the negative regularity Besov scale, with regularity arbitrarly close to $-1$. Our results rely on turning a priori bounds for the solution in negative Besov spaces into bounds in the positive regularity scale.},
author = {Chemin, Jean-Yves, Planchon, Fabrice},
journal = {Bulletin de la Société Mathématique de France},
keywords = {Navier-Stokes equations; blow-up criterion; Besov spaces},
language = {eng},
number = {4},
pages = {583-597},
publisher = {Société mathématique de France},
title = {Self-improving bounds for the Navier-Stokes equations},
url = {http://eudml.org/doc/272604},
volume = {140},
year = {2012},
}

TY - JOUR
AU - Chemin, Jean-Yves
AU - Planchon, Fabrice
TI - Self-improving bounds for the Navier-Stokes equations
JO - Bulletin de la Société Mathématique de France
PY - 2012
PB - Société mathématique de France
VL - 140
IS - 4
SP - 583
EP - 597
AB - We consider regular solutions to the Navier-Stokes equation and provide an extension to the Escauriaza-Seregin-Sverak blow-up criterion in the negative regularity Besov scale, with regularity arbitrarly close to $-1$. Our results rely on turning a priori bounds for the solution in negative Besov spaces into bounds in the positive regularity scale.
LA - eng
KW - Navier-Stokes equations; blow-up criterion; Besov spaces
UR - http://eudml.org/doc/272604
ER -

References

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  1. [1] M. Cannone – « A generalization of a theorem by Kato on Navier-Stokes equations », Rev. Mat. Iberoamericana13 (1997), p. 515–541. Zbl0897.35061MR1617394
  2. [2] M. Cannone & F. Planchon – « On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations », Rev. Mat. Iberoamericana16 (2000), p. 1–16. Zbl0965.35121MR1768531
  3. [3] J.-Y. Chemin – « Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel », J. Anal. Math.77 (1999), p. 27–50. Zbl0938.35125MR1753481
  4. [4] A. Cheskidov & R. Shvydkoy – « The regularity of weak solutions of the 3D Navier-Stokes equations in B ∞ , ∞ - 1 », Arch. Ration. Mech. Anal.195 (2010), p. 159–169. Zbl1186.35137MR2564471
  5. [5] I. Gallagher, D. Iftimie & F. Planchon – « Asymptotics and stability for global solutions to the Navier-Stokes equations », Ann. Inst. Fourier (Grenoble) 53 (2003), p. 1387–1424. Zbl1038.35054MR2032938
  6. [6] I. Gallagher, G. Koch & F. Planchon – « A profile decomposition approach to the L t ∞ ( L x 3 ) Navier-Stokes regularity criterion », Math. Ann. (2012), doi:10.1007/s00208-012-0830-0 and arXiv:math/1012.0145. Zbl1291.35180
  7. [7] L. Iskauriaza, G. A. Serëgin & V. Shverak – « L 3 , ∞ -solutions of Navier-Stokes equations and backward uniqueness », Uspekhi Mat. Nauk58 (2003), p. 3–44. Zbl1064.35134MR1992563
  8. [8] T. Kato – « Strong L p -solutions of the Navier-Stokes equation in 𝐑 m , with applications to weak solutions », Math. Z.187 (1984), p. 471–480. Zbl0545.35073MR760047
  9. [9] T. Kato & H. Fujita – « On the nonstationary Navier-Stokes system », Rend. Sem. Mat. Univ. Padova32 (1962), p. 243–260. Zbl0114.05002MR142928
  10. [10] H. Koch & D. Tataru – « Well-posedness for the Navier-Stokes equations », Adv. Math.157 (2001), p. 22–35. Zbl0972.35084MR1808843
  11. [11] J. Leray – « Essai sur le mouvement d’un liquide visqueux emplissant l’espace », Acta Mathematica63 (1933), p. 193–248. JFM60.0726.05
  12. [12] F. Planchon – « Asymptotic behavior of global solutions to the Navier-Stokes equations in 𝐑 3 », Rev. Mat. Iberoamericana14 (1998), p. 71–93. Zbl0910.35096MR1639283

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