Advertisement

Archive for Rational Mechanics and Analysis

, Volume 230, Issue 2, pp 641–663 | Cite as

On the Local Type I Conditions for the 3D Euler Equations

  • Dongho Chae
  • Jörg Wolf
Article
  • 155 Downloads

Abstract

We prove local non blow-up theorems for the 3D incompressible Euler equations under local Type I conditions. More specifically, for a classical solution \({v\in L^\infty (-1,0; L^2 ( B(x_0,r)))\cap L^\infty_{{\rm loc}} (-1,0; W^{1, \infty} (B(x_0, r)))}\) of the 3D Euler equations, where \({B(x_0,r)}\) is the ball with radius r and the center at x0, if the limiting values of certain scale invariant quantities for a solution v(·, t) as \({t\to 0}\) are small enough, then \({ \nabla v(\cdot,t) }\) does not blow-up at t = 0 in B(x0, r).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beale J.T., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys. 94, 61–66 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Caffarelli L., Kohn R., Nirenberg L.: Partial Regularity of suitable weak solutions of the Navier–Stokes equations. Comm. Pure Appl. Math. 35, 771–831 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chae D.: On the generalized self-similar singularities for the Euler and the Navier–Stokes equations. J. Funct. Anal. 258(9), 2865–2883 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chae, D., Wolf, J.: Energy concentrations and Type I blow-up for the 3D Euler equations arXiv:1706.02020
  5. 5.
    Chae, D., Wolf, J.: Local regularity criterion of the Beale-Kato-Majda type for the 3D Euler equations, arXiv:1711.06415
  6. 6.
    Chen C.-C., Strain R.M., Tsai T.-P., Yau H.-T.: Lower Bounds on the Blow-Up Rate of the Axisymmetric Navier–Stokes Equations II. Comm. P.D.E. 34, 203–232 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Constantin P.: On the Euler equations of incompressible fluids. Bull. Amer. Math. Soc. 44(4), 603–621 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Constantin P., Fefferman C., Majda A.: Geometric constraints on potential singularity formulation in the 3-D Euler equations. Comm. P.D.E. 21(3–4), 559–571 (1996)zbMATHGoogle Scholar
  9. 9.
    Deng J., Hou T.Y., Yu X.: Improved geometric conditions for non-blow up of the 3D incompressible Euler equations. Comm. P.D.E. 31(1–3), 293–306 (2006)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hou T.Y., Li R.: Nonexistence of local self-similar blow-up for the 3D incompressible Navier–Stokes equations. Discrete Contin. Dyn. Syst. 18, 637–642 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kato T., Ponce G.: Commutator estimates and the Euler and Navier–Stokes equations. Comm. Pure Appl. Math. 41(7), 891–907 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kozono H., Taniuchi Y.: Limiting case of the Sobolev inequality in BMO, with applications to the Euler equations. Comm. Math. Phys. 214, 191–200 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Majda, A., Bertozzi, A.: Vorticity and incompressible flow, Cambridge Univ. Press, 2002Google Scholar
  14. 14.
    Seregin G., Šverák V.: On Type I singularities of the local axially symmetric solutions of the Navier–Stokes equations. Comm. P.D.E. 34, 171–201 (2009)CrossRefzbMATHGoogle Scholar
  15. 15.
    Simon J.: Compact sets in the space L p (0, T; B). Ann. Mat. Pura Appl. 146, 65–96 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Stein, E.M.: Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970Google Scholar
  17. 17.
    Wolf J.: On the local pressure of the Navier–Stokes equations and related systems. Adv. Differ. Equ. 22, 305–338 (2017)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsChung-Ang UniversitySeoulRepublic of Korea

Personalised recommendations