Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Finite time blow up for a Navier-Stokes like equation


Author: Stephen Montgomery-Smith
Journal: Proc. Amer. Math. Soc. 129 (2001), 3025-3029
MSC (2000): Primary 35Q30, 46E35; Secondary 34G20, 37L05, 47D06, 47H10
Published electronically: April 17, 2001
MathSciNet review: 1840108
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider an equation similar to the Navier-Stokes equation. We show that there is initial data that exists in every Triebel-Lizorkin or Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the solution is in no Triebel-Lizorkin or Besov space (and hence in no Lebesgue or Sobolev space). The purpose is to show the limitations of the so-called semigroup method for the Navier-Stokes equation. We also consider the possibility of existence of solutions with initial data in the Besov space $\dot B^{-1,\infty}_\infty$. We give initial data in this space for which there is no reasonable solution for the Navier-Stokes like equation.


References [Enhancements On Off] (What's this?)

  • [Ca1] Marco Cannone, Ondelettes, paraproduits et Navier-Stokes, Diderot Editeur, Paris, 1995 (French). With a preface by Yves Meyer. MR 1688096
  • [Ca2] Cannone, Marco, Rôle des oscillations et des espaces de Besov dans la résolution des équations de Navier-Stokes. Document de synthèse présenté pour obtenir une Habilitation à diriger des Recherches, Université de Paris 7, 1-35, 1999.
  • [FJW] Michael Frazier, Björn Jawerth, and Guido Weiss, Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics, vol. 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991. MR 1107300
  • [FK] Hiroshi Fujita and Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269–315. MR 0166499
  • [FLT] Giulia Furioli, Pierre-Gilles Lemarié-Rieusset, and Elide Terraneo, Sur l’unicité dans 𝐿³(𝑅³) des solutions “mild” des équations de Navier-Stokes, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 12, 1253–1256 (French, with English and French summaries). MR 1490408, 10.1016/S0764-4442(97)82348-8
  • [GM] Yoshikazu Giga and Tetsuro Miyakawa, Navier-Stokes flow in 𝐑³ with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations 14 (1989), no. 5, 577–618. MR 993821, 10.1080/03605308908820621
  • [K] Tosio Kato, Strong 𝐿^{𝑝}-solutions of the Navier-Stokes equation in 𝑅^{𝑚}, with applications to weak solutions, Math. Z. 187 (1984), no. 4, 471–480. MR 760047, 10.1007/BF01174182
  • [KF] Tosio Kato and Hiroshi Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Mat. Univ. Padova 32 (1962), 243–260. MR 0142928
  • [KT] Koch, Herbert; Tataru, Daniel, Well-posedness for the Navier-Stokes equation, preprint.
  • [O] Oru F., Rôle des oscillations dans quelques problèmes d'analyse non linéaire. Thèse de Doctorat de l'ENS Cachan, 1998.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35Q30, 46E35, 34G20, 37L05, 47D06, 47H10

Retrieve articles in all journals with MSC (2000): 35Q30, 46E35, 34G20, 37L05, 47D06, 47H10


Additional Information

Stephen Montgomery-Smith
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: stephen@math.missouri.edu

DOI: https://doi.org/10.1090/S0002-9939-01-06062-2
Keywords: Navier-Stokes equation, semigroup, fixed point method, Triebel-Lizorkin space, Besov space
Received by editor(s): March 1, 2000
Published electronically: April 17, 2001
Additional Notes: The author was partially supported by NSF grant DMS 9870026.
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society