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On the Partial Regularity of a 3D Model of the Navier-Stokes Equations

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Abstract

We study the partial regularity of a 3D model of the incompressible Navier-Stokes equations which was recently introduced by the authors in [11]. This model is derived for axisymmetric flows with swirl using a set of new variables. It preserves almost all the properties of the full 3D Euler or Navier-Stokes equations except for the convection term which is neglected in the model. If we add the convection term back to our model, we would recover the full Navier-Stokes equations. In [11], we presented numerical evidence which seems to support that the 3D model develops finite time singularities while the corresponding solution of the 3D Navier-Stokes equations remains smooth. This suggests that the convection term play an essential role in stabilizing the nonlinear vortex stretching term. In this paper, we prove that for any suitable weak solution of the 3D model in an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero. The partial regularity result of this paper is an analogue of the Caffarelli-Kohn-Nirenberg theory for the 3D Navier-Stokes equations.

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References

  1. Beale J.T., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94(1), 61–66 (1984)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Cao C., Titi E.S.: Global well-posedness of the three-dimensional primitive equations of large scale ocean and atmosphere dynamics. Ann. Math. 166(1), 245–267 (2007)

    MATH  MathSciNet  Google Scholar 

  4. Chen, C., Strain, R.M., Tsai, T., Yau, H.T.: Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations. Int. Math. Res. Not. 2008 (2008), published on March 14, 2008, doi:10.1093/imrn/rnn016

  5. Chen, C., Strain, R.M., Tsai, T., Yau, H.T.: Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations II. http://arXiv.org/abs/070/796v1 [Math.AP], 2007

  6. Deng J., Hou T.Y., Yu X.: Geometric properties and non-blowup of 3-D incompressible Euler flow. Commun. PDEs 30, 225–243 (2005)

    MATH  MathSciNet  Google Scholar 

  7. Deng J., Hou T.Y., Yu X.: Improved geometric conditions for non-blowup of 3D incompressible Euler equation. Commun. PDEs 31, 293–306 (2006)

    MATH  MathSciNet  Google Scholar 

  8. Iskauriaza, L., Seregin, G.A., Shverak, V.: L 3,∞-solutions of Navier-Stokes equations and backward uniqueness. (Russian) Usp. Mat. Nauk 582(350), 3–44 (2003); translation in Russ. Math. Surveys 58(2) 211–250 (2003)

  9. Fefferman, C.: http://www.claymath.org/millennium/Navier-Stokes_Equations

  10. Hou T.Y., Li C.: Dynamic stability of the 3D axi-symmetric Navier-Stokes equations with swirl. CPAM 61(5), 661–697 (2008)

    MATH  MathSciNet  Google Scholar 

  11. Hou, T.Y., Lei, Z.: On the stabilizing effect of convection in 3D incompressible flows. CPAM, published online on June 10, 2008, doi:10.1002/cpa.20254

  12. Hou T.Y., Lei Z., Li C.: Global regularity of the 3D Axi-symmetric Navier-Stokes equations with anisotropic data. Commun. PDEs 33(9), 1622–1637 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hopf E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951)

    MATH  MathSciNet  Google Scholar 

  14. Kang K.: Regularity of axially symmetric flows in a half-space in three dimensions. SIAM J. Math. Anal. 35(6), 1636–1643 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Koch, G., Nadirashvili, N., Seregin, G., Sverak, V.: Liouville theorems for the Navier-Stokes equations and applications. http://arXiv.org/abs/0709.3599.vl [math.AP], 2007

  16. Kozono H., Taniuchi Y.: Bilinear estimates in BMO and the Navier-Stokes equations. Math. Z. 235(1), 173–194 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ladyzhenskay, O.A.: On uniqueness and smoothness of generalized solutions to the Navier-Stokes equations. Zap. Nauv̌cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5, 169–185 (1967); English transl., Sem. Math. V. A. Steklov Math. Inst. Leningrad 5, 60–66 (1969)

  18. Ladyzhenskaya O.A.: Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry. (Russian) Zap. Nauv̌cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7, 155–177 (1968)

    MATH  Google Scholar 

  19. Ladyzhenskaya O.A., Seregin G.A.: On partial regularity of suitable weak solutions to the three- dimensional Navier-Stokes equations. (English summary) J Math. Fluid Mech. 1(4), 356–387 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Leonardi S., Málek J., Nev̌cas J., Pokorný M.: On axially symmetric flows in R 3. Z. Anal. Anwendungen 18(3), 639–649 (1999)

    MATH  MathSciNet  Google Scholar 

  21. Leray J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta. Math. 63, 193–248 (1934)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lin F.-h.: A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math. 51(3), 241–257 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  23. Liu J.G., Wang W.C.: Convergence analysis of the energy and helicity preserving scheme for axisymmetric flows. SINUM 44(6), 2456–2480 (2006)

    MATH  Google Scholar 

  24. Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics 27. Cambridge: Cambridge University Press, 2002

  25. Okamoto H., Ohkitani K.: On the role of the convection term in the equations of motion of incompressible fluid. J. Phys. Soc. Japan 74(10), 2737–2742 (2005)

    Article  MATH  ADS  Google Scholar 

  26. Prodi G.: Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48(4), 173–182 (1959)

    MATH  MathSciNet  Google Scholar 

  27. Scheffer V.: Partial regularity of solutions to the Navier-Stokes equations. Pac. J. Math. 66(2), 535–552 (1976)

    MATH  MathSciNet  Google Scholar 

  28. Scheffer V.: Hausdorff measure and the Navier-Stokes equations. Commun. Math. Phys. 55(2), 97–112 (1977)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. Scheffer V.: The Navier-Stokes equations on a bounded domain. Commun. Math. Phys. 73(1), 1–42 (1980)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. Scheffer V.: Boundary regularity for the Navier-Stokes equations. Commun. Math. Phys. 85(2), 275–299 (1982)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. Serrin J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal. 9, 187–195 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  32. Serrin, J.: The initial value problem for the Navier-Stokes equations. In: Nonlinear Problems. Madison WI: Univ. of Wisconsin Press, pp. 69–98, 1963

  33. Struwe M.: On partial regularity results for the Navier-Stokes equations. Comm. Pure Appl. Math. 41(4), 437–458 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  34. Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis. Providence RI: AMS Chelsea Publishing, 2001

  35. Ukhovskii, M.R., Iudovich, V.I.: Axially symmetric flows of ideal and viscous fluids filling the whole space. Prikl. Mat. Meh. 32, 59–69 (1968) (Russian); translated as J. Appl. Math. Mech. 32, 52–61 (1968)

  36. Beirãoda Veiga, H.: A new regularity class for the Navier-Stokes equations in R n. (English summary) A Chinese summary appears in Chinese Ann. Math. Ser. A 16(6), (1995). 797, Chinese Ann. Math. Ser. B 16(4), 407–412 (1995)

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Authors and Affiliations

  1. Applied and Comput. Math, Caltech, Pasadena, CA, 91125, USA

    Thomas Y. Hou

  2. School of Mathematical Sciences, Fudan University, Shanghai, 200433, P.R. China

    Zhen Lei

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  1. Thomas Y. Hou
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  2. Zhen Lei
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Correspondence to Thomas Y. Hou.

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Communicated by H.-T. Yau

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Hou, T.Y., Lei, Z. On the Partial Regularity of a 3D Model of the Navier-Stokes Equations. Commun. Math. Phys. 287, 589–612 (2009). https://doi.org/10.1007/s00220-008-0689-9

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