Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
권호 표지
DOI QR코드

DOI QR Code

CRITERION FOR BLOW-UP IN THE EULER EQUATIONS VIA CERTAIN PHYSICAL QUANTITIES

  • Kim, Namkwon (DEPARTMENT OF MATHEMATICS, CHOSUN UNIVERSITY)
  • Received : 2012.11.19
  • Accepted : 2012.12.07
  • Published : 2012.12.25
Download PDF
⟨ Previous Next ⟩

Abstract

We consider the (possible) finite time blow-up of the smooth solutions of the 3D incompressible Euler equations in a smooth domain or in $R^3$. We derive blow-up criteria in terms of $L^{\infty}$ of the partial component of Hessian of the pressure together with partial component of the vorticity.

Keywords

References

  1. H. Bae, B. Jin, and N. Kim, On the localized blow-up of the 3D incompressible Euler equations, preprint, 2012
  2. Beale J T, Kato T and Majda A 1984, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94, no. 1, pp. 61-66().
  3. D. Chae, Incompressible Euler Equations: The Blow-up Problem and Related Results, in Handbook of Differential Equations: Evolutionary Equations, Elsevier, Volume 4, pp. 1-55, 2008.
  4. D.Chae, On the finite-time singularities of the 3D incompressible Euler equations, Comm.Pure.Appl.Math.Vol.LX, 0597-0617(2007).
  5. D. Chae, Remarks on the blow-up criterion of the three-dimensional Euler equations, Nonlinearity 18, no. 3, 1021-1029(2005). https://doi.org/10.1088/0951-7715/18/3/005
  6. P. Constantin, Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations, Comm. Math. Phys. 104, pp. 311-326(1986). https://doi.org/10.1007/BF01211598
  7. T. Kato, Nonstationary flows of viscous and ideal fluids in R, J. Func. Anal., 9, pp. 296-305(1972). https://doi.org/10.1016/0022-1236(72)90003-1
  8. N. Kim, Remarks on the blow-up of solutions for the 3-D Euler equations, Differential and Integral equations, 14, no.2,129-140(2001).
  9. G. Ponce, Remarks on a paper: "Remarks on the breakdown of smooth solutions for the 3-D Euler equations" Comm. Math. Phys. 94, no. 1, 61-66(1984) by J. T. Beale, T. Kato and A. Majda., Comm. Math. Phys. 98, pp. 349-353 (1985). https://doi.org/10.1007/BF01205787
  10. R. Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal., 20(1975), pp. 32-43. https://doi.org/10.1016/0022-1236(75)90052-X