Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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REMARK ON PARTICLE TRAJECTORY FLOWS WITH UNBOUNDED VORTICITY

  • Received : 2014.08.25
  • Accepted : 2014.10.10
  • Published : 2014.11.15
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Abstract

The existence and the regularity of the particle trajectory flow X(x, t) along a velocity field u on $\mathbb{R}^n$ are discussed under the BMO-blow-up condition: $${\int}_{0}^{T}{\parallel}{\omega}({\tau}){\parallel}_{BMO}d{\tau}<{\infty}$$ of the vorticity ${\omega}{\equiv}{\nabla}{\times}u$. A comment on our result related with the mystery of turbulence is presented.

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References

  1. J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1984), 61-66. https://doi.org/10.1007/BF01212349
  2. D. Chae, On the Well-Posedness of the Euler Equations in the Triebel-Lizorkin Spaces, Comm. Pure Appl. Math. 55 (2002), no. 5, 654-678. https://doi.org/10.1002/cpa.10029
  3. J.-Y. Chemin, Perfect incompressible fluids, Clarendon Press, Oxford, 1981.
  4. E. A. Coddington and N. Levinson, Theory Ordinary Differential Equations, McGraw-Hill, New York, 1955.
  5. P. Constantin, Geometric statistics in turbulence, SIAM Rev. 36 (1994), 73-98. https://doi.org/10.1137/1036004
  6. P. Constantin, C. Fefferman, and A. Majda, Geometric constraints on potentially singular solutions for the 3D Euler equations, Commun. Part. Diff. Eq. 21 (1996), 559-571. https://doi.org/10.1080/03605309608821197
  7. H. Kozono and Y. Taniuchi, Limiting case of Sobolev inequality in BMO, Commun. Math. Phys. 214 (2000), 191-200. https://doi.org/10.1007/s002200000267
  8. H. Kozono, T. Ogawa, and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z. 242 (2002), 251-278. https://doi.org/10.1007/s002090100332
  9. P. Hartman, Ordinary Differential Equations 2nd Ed., Society for Industrial and Applied Mathematics, Philadelphia, 2002.
  10. K. Ohkitani, A miscellany of basic issues on incompressible fluid equations, Nonlinearity 21 (2008), 255-271. https://doi.org/10.1088/0951-7715/21/12/T02
  11. H. Pak and Y. Park, Existence of solution for the Euler equations in a critical Besov space $B^1_{{\infty},1}(\mathbb{R}^n)$, Commun. Part. Differ. Eq. 29 (2004), 1149-1166. https://doi.org/10.1081/PDE-200033764
  12. G. Ponce, Remarks on a paper by J. T. Beale, T. Kato and A. Majda, Comm. Math. Phys. 98 (1985), 349-353. https://doi.org/10.1007/BF01205787
  13. A. Pumir and E. D. Siggia, Collapsing solutions to the 3-D Euler equations, Phys. Fluids A 2 (1990), 220-241. https://doi.org/10.1063/1.857824
  14. E. M. Stein, Harmonic analysis; Real-variable methods, orthogonality, and osillatory integrals, Princeton Mathematical Series, 43, 1993.
  15. H. Triebel, Theory of Function spaces, Birkhauser, 1983.
  16. M. Vishik, Hydrodynamics in Besov spaces, Arch. Rational Mech. Anal. 145 (1998), 197-214. https://doi.org/10.1007/s002050050128
  17. M. Vishik, Incompressible flows of an ideal fluid with unbounded vorticity, Comm. Math. Phys. 213 (2000), 697-731. https://doi.org/10.1007/s002200000255
  18. V. I. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math Res. Lett. 2 (1995), 27-38. https://doi.org/10.4310/MRL.1995.v2.n1.a4

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