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)

 

 

Some elementary estimates for the Navier-Stokes system


Author: Jean Cortissoz
Journal: Proc. Amer. Math. Soc. 137 (2009), 3343-3353
MSC (2000): Primary 35Q30
Published electronically: May 29, 2009
MathSciNet review: 2515404
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the incompressible Navier-Stokes equations in $ {\mathbb{T}}^3=[0,1]^3$ with periodic boundary conditions. We show that a weak solution of the Navier-Stokes equations that is small in $ L^{\infty}(0,T;\Phi(2))$ is also smooth. We also give elementary proofs of some classical regularity results for the Navier-Stokes equations involving the Sobolev space $ H^{\frac{1}{2}}({\mathbb{T}}^3)$.


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

  • [1] Ya. G. Sinai and M. D. Arnold, Global existence and uniqueness theorem for 3D-Navier-Stokes system on 𝕋³ for small initial conditions in the spaces Φ(𝛼), Pure Appl. Math. Q. 4 (2008), no. 1, Special Issue: In honor of Grigory Margulis., 71–79. MR 2405995, 10.4310/PAMQ.2008.v4.n1.a2
  • [2] L. Iskauriaza, G. A. Serëgin, and V. Shverak, 𝐿_{3,∞}-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk 58 (2003), no. 2(350), 3–44 (Russian, with Russian summary); English transl., Russian Math. Surveys 58 (2003), no. 2, 211–250. MR 1992563, 10.1070/RM2003v058n02ABEH000609
  • [3] A. Cheskidov, R. Shvydkoy, On the regularity of weak solutions of the 3D Navier-Stokes equations in $ B_{\infty,\infty}^{-1}$, arXiv:0708.3067v2.
  • [4] Yoshikazu Giga, Solutions for semilinear parabolic equations in 𝐿^{𝑝} and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations 62 (1986), no. 2, 186–212. MR 833416, 10.1016/0022-0396(86)90096-3
  • [5] V. I and Ya. G. Sinaĭ, New results in mathematical and statistical hydrodynamics, Uspekhi Mat. Nauk 55 (2000), no. 4(334), 25–58 (Russian, with Russian summary); English transl., Russian Math. Surveys 55 (2000), no. 4, 635–666. MR 1786729, 10.1070/rm2000v055n04ABEH000313
  • [6] Hideo Kozono and Hermann Sohr, Regularity criterion of weak solutions to the Navier-Stokes equations, Adv. Differential Equations 2 (1997), no. 4, 535–554. MR 1441855
  • [7] Jean Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248 (French). MR 1555394, 10.1007/BF02547354
  • [8] J. C. Mattingly and Ya. G. Sinai, An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equations, Commun. Contemp. Math. 1 (1999), no. 4, 497–516. MR 1719695, 10.1142/S0219199799000183
  • [9] Roger Temam, Navier-Stokes equations and nonlinear functional analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 41, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1983. MR 764933
  • [10] Wolf von Wahl, Regularity of weak solutions of the Navier-Stokes equations, Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 497–503. MR 843635

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35Q30

Retrieve articles in all journals with MSC (2000): 35Q30


Additional Information

Jean Cortissoz
Affiliation: Departamento de Matemáticas, Universidad de Los Andes, Bogotá DC, Colombia
Email: jcortiss@uniandes.edu.co

DOI: https://doi.org/10.1090/S0002-9939-09-09989-4
Keywords: Navier-Stokes equations, regularity
Received by editor(s): October 14, 2008
Published electronically: May 29, 2009
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.