Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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GLOBAL EXISTENCE FOR A PARTIALLY LINEAR 3D EULER FLOW

  • Received : 2017.03.09
  • Accepted : 2017.10.25
  • Published : 2018.01.01
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Abstract

We consider a certain three dimensional Euler flow with infinite energy, which is sometimes called the columnar or two and half dimensional flow. We prove the global smoothness of such flow in ${\mathbb{R}}^3$ when the initial data is in some Sobolev or Besov spaces and ${\partial}_3u_3$ is nonnegative.

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References

  1. H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011.
  2. M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, in Handbook of mathematical fluid dynamics. Vol. III, 161-244, North-Holland, Amsterdam.
  3. D. Chae, Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. Anal. 38 (2004), no. 3-4, 339-358.
  4. P. Constantin, The Euler equations and nonlocal conservative Riccati equations, Internat. Math. Res. Notices 2000, no. 9, 455-465. https://doi.org/10.1155/S1073792800000258
  5. R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, November 14, 2005.
  6. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
  7. J. D. Gibbon, A. S. Fokas, and C. R. Doering, Dynamically stretched vortices as solutions of the 3D Navier-Stokes equations, Phys. D 132 (1999), no. 4, 497-510. https://doi.org/10.1016/S0167-2789(99)00067-6
  8. J. D. Gibbon, D. R. Moore, and J. T. Stuart, Exact, infinite energy, blow-up solutions of the three-dimensional Euler equations, Nonlinearity 16 (2003), no. 5, 1823-1831. https://doi.org/10.1088/0951-7715/16/5/315
  9. W. Hardle, G. Kerkyacharian, D. Picard, and A. Tsybakov, Wavelets, Approximation, and Statistical Applications, Lecture Notes in Statistics, 129, Springer-Verlag, New York, 1998.
  10. T. Kato, Nonstationary flows of viscous and ideal fluids in ${\mathbb{R}}^3$, J. Functional Analysis 9 (1972), 296-305. https://doi.org/10.1016/0022-1236(72)90003-1
  11. T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891-907. https://doi.org/10.1002/cpa.3160410704
  12. N. Kim and B. Lkhagvasuren, On the global existence of columnar solutions of the Navier-Stokes equations, submitted.
  13. P. G. Lemarie-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431, Chapman & Hall/CRC, Boca Raton, FL, 2002.
  14. A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, Cambridge, 2002.
  15. Y. Meyer, Wavelets and Operators, translated from the 1990 French original by D. H. Salinger, Cambridge Studies in Advanced Mathematics, 37, Cambridge University Press, Cambridge, 1992.
  16. K. Ohkitani and J. D. Gibbon, Numerical study of singularity formation in a class of Euler and Navier-Stokes flows, Phys. Fluids 12 (2000), no. 12, 3181-3194. https://doi.org/10.1063/1.1321256
  17. H. Triebel, A note on wavelet bases in function spaces, in Orlicz centenary volume, 193-206, Banach Center Publ., 64, Polish Acad. Sci. Inst. Math., Warsaw, 2004.