대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제35권1호
  • /
  • Pages.339-345
  • /
  • 2020
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

ON THE IMPROVED REGULARITY CRITERION OF THE SOLUTIONS TO THE NAVIER-STOKES EQUATIONS

  • Gala, Sadek (Department of Sciences Exactes Ecole Normale Superieure of Mostaganem)
  • 투고 : 2019.01.17
  • 심사 : 2019.05.09
  • 발행 : 2020.01.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

This note deals with the question of the regularity of (Leray) weak solutions of the Navier-Stokes equations in terms of the pressure. This criterion improves on the existing results.

키워드

참고문헌

  1. C. H. Chan and A. Vasseur, Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations, Methods Appl. Anal. 14 (2007), no. 2, 197-212. https://doi.org/10.4310/MAA.2007.v14.n2.a5
  2. B. Dong, G. Sadek, and Z. Chen, On the regularity criteria of the 3D Navier-Stokes equations in critical spaces, Acta Math. Sci. Ser. B (Engl. Ed.) 31 (2011), no. 2, 591-600. https://doi.org/10.1016/S0252-9602(11)60259-2
  3. J. Fan, S. Jiang, and G. Nakamura, On logarithmically improved regularity criteria for the Navier-Stokes equations in ${\mathbb{R}^n}$, IMA J. Appl. Math. 76 (2011), no. 2, 298-311. https://doi.org/10.1093/imamat/hxq035
  4. J. Fan, S. Jiang, G. Nakamura, and Y. Zhou, Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations, J. Math. Fluid Mech. 13 (2011), no. 4, 557-571. https://doi.org/10.1007/s00021-010-0039-5
  5. J. Fan, S. Jiang, and G. Ni, On regularity criteria for the n-dimensional Navier-Stokes equations in terms of the pressure, J. Differential Equations 244 (2008), no. 11, 2963-2979. https://doi.org/10.1016/j.jde.2008.02.030
  6. J. Fan and T. Ozawa, Regularity criterion for weak solutions to the Navier-Stokes equations in terms of the gradient of the pressure, J. Inequal. Appl. 2008 (2008), Art. ID 412678, 6 pp. https://doi.org/10.1155/2008/412678
  7. J. Fan and T. Ozawa, Logarithmically improved regularity criteria for Navier-Stokes and related equations, Math. Methods Appl. Sci. 32 (2009), no. 17, 2309-2318. https://doi.org/10.1002/mma.1140
  8. S. Gala, Regularity criterion on weak solutions to the Navier-Stokes equations, J. Korean Math. Soc. 45 (2008), no. 2, 537-558. https://doi.org/10.4134/JKMS.2008.45.2.537
  9. S. Gala, Remark on a regularity criterion in terms of pressure for the Navier-Stokes equations, Quart. Appl. Math. 69 (2011), no. 1, 147-155. https://doi.org/10.1090/S0033-569X-2011-01206-0
  10. S. Gala, A remark on the blow-up criterion of strong solutions to the Navier-Stokes equations, Appl. Math. Comput. 217 (2011), no. 22, 9488-9491. https://doi.org/10.1016/j.amc.2011.03.156
  11. S. Gala, Remarks on regularity criterion for weak solutions to the Navier-Stokes equations in terms of the gradient of the pressure, Appl. Anal. 92 (2013), no. 1, 96-103. https://doi.org/10.1080/00036811.2011.593172
  12. P. Gerard, Y. Meyer, and F. Oru, Inegalites de Sobolev precisees, in Seminaire sur les Equations aux Derivees Partielles, 1996-1997, Exp. IV, 11 pp, Ecole Polytech., Palaiseau, 1997.
  13. X. He and S. Gala, Regularity criterion for weak solutions to the Navier-Stokes equations in terms of the pressure in the class $L^{2}(0,T;\;\dot{B}_{{\infty}}^{-1},\;_{{\infty}}(\mathbb{R}^{3}))$, Nonlinear Anal. 12 (2011), 3602-3607. https://doi.org/10.1016/j.nonrwa.2011.06.018
  14. E. Hopf, Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213-231. https://doi.org/10.1002/mana.3210040121
  15. J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), no. 1, 193-248. https://doi.org/10.1007/BF02547354
  16. J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), 187-195. https://doi.org/10.1007/BF00253344
  17. M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 4, 437-458. https://doi.org/10.1002/cpa.3160410404
  18. Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in ${\mathbb{R}^3}$, Proc. Amer. Math. Soc. 134 (2006), no. 1, 149-156. https://doi.org/10.1090/S0002-9939-05-08312-7
  19. Y. Zhou, On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in $\mathbb{R}^N$, Z. Angew. Math. Phys. 57 (2006), no. 3, 384-392. https://doi.org/10.1007/s00033-005-0021-x
  20. Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math. 24 (2012), no. 4, 691-708. https://doi.org/10.1515/form.2011.079
  21. Y. Zhou and S. Gala, Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces, J. Math. Anal. Appl. 356 (2009), no. 2, 498-501. https://doi.org/10.1016/j.jmaa.2009.03.038
  22. Y. Zhou and S. Gala, Regularity criteria in terms of the pressure for the Navier-Stokes equations in the critical Morrey-Campanato space, Z. Anal. Anwend. 30 (2011), no. 1, 83-93. https://doi.org/10.4171/ZAA/1425
  23. Y. Zhou and S. Gala, Remarks on logarithmical regularity criteria for the Navier-Stokes equations, J. Math. Phys. 52 (2011), no. 6, 063503, 9 pp. https://doi.org/10.1063/1.3569967
  24. Y. Zhou and Z. Lei, Logarithmically improved criteria for Euler and Navier-Stokes equations, Commun. Pure Appl. Anal. 12 (2013), no. 6, 2715-2719. https://doi.org/10.3934/cpaa.2013.12.2715