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Interior Regularity Criteria for Suitable Weak Solutions of the Navier-Stokes Equations

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Abstract

We present new interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations: a suitable weak solution is regular near an interior point z if either the scaled \({L^{p,q}_{x,t}}\) -norm of the velocity with 3/p + 2/q ≤ 2, 1 ≤ q ≤ ∞, or the \({L^{p,q}_{x,t}}\) -norm of the vorticity with 3/p + 2/q ≤ 3, 1 ≤  q <  ∞, or the \({L^{p,q}_{x,t}}\) -norm of the gradient of the vorticity with 3/p + 2/q ≤ 4, 1 ≤  q, 1 ≤  p, is sufficiently small near z.

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References

  1. Beirão da Veiga H. (1995). A new regularity class for the Navier-Stokes equations in \({\mathbb{R}^{n}}\) Chin. Ann. of Math. 16(B, 4): 407–412

    MATH  Google Scholar 

  2. Caffarelli L., Kohn R. and Nirenberg L. (1982). Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35: 771–831

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Chae, D., Kang, K., Lee, J.: On the interior regularity of suitable weak solutions to the Navier-Stokes equations. Comm. Partial Differ. Eqs. (accepted)

  4. Chen Z.-M. and Price W.G. (2001). Blow-up rate estimates for weak solutions of the Navier-Stokes equations. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457(2015): 2625–2642

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Choe H. and Lewis J.L. (2000). On the singular set in the Navier-Stokes equations. J. Funct. Anal. 175(2): 348–369

    Article  MATH  MathSciNet  Google Scholar 

  6. Constantin P. (1990). Navier-Stokes equations and Area of Interfaces. Commun. Math. Phys. 129: 241–266

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Escauriaza, L., Seregin, G., Šverák, V.: L 3,∞-solutions of Navier-Stokes equations and backward uniqueness. Usp. Mat. Nauk 58 (2(350)) 3–44 (2003) (in Russian); translation from Russ. Math. Surv. 58(2), 211–250 (2003)

    Google Scholar 

  8. Fabes E.B., Jones B.F. and Riviére N.M. (1972). The initial value problem for the Navier-Stokes equations with data in L p. Arch. Ration. Mech. Anal. 45: 222–240

    Article  MATH  Google Scholar 

  9. Giga Y. (1986). Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system. J. Differ Eqs. 62(2): 186–212

    Article  MathSciNet  Google Scholar 

  10. Gustafson S., Kang K. and Tsai T.-P. (2006). Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary. J. Differ. Eqs. 226: 594–618

    Article  MATH  MathSciNet  Google Scholar 

  11. Hopf E. (1951). Über die Anfangswertaufgabe fr die hydrodynamischen Grundgleichungen. Math. Nachr. 4: 213–231

    MATH  MathSciNet  Google Scholar 

  12. Kang K. (2004). On boundary regularity of the Navier-Stokes equations. Comm. Partial Differ. Eqs. 29(7–8): 955–987

    Article  MATH  Google Scholar 

  13. Kozono H. (1998). Removable singularities of weak solutions to the Navier-Stokes equations. Comm. Partial Differ. Eqs. 23(5–6): 949–966

    MATH  MathSciNet  Google Scholar 

  14. Kozono H. (2001). Weak solutions of the Navier-Stokes equations with test functions in the weak-L n space. Tohoku Math J. (2) 53: 55–79

    MATH  MathSciNet  Google Scholar 

  15. Ladyzenskaja O.A. (1967). Uniqueness and smoothness of generalized solutions of Navier-Stokes equations, (Russian) Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5: 169–185

    MathSciNet  Google Scholar 

  16. Ladyzenskaja O.A. and Seregin G.A. (1999). On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. J. Math. Fluid Mech. 1: 356–387

    Article  ADS  MathSciNet  Google Scholar 

  17. Leray J. (1934). Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 64: 193–248

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Nečas J., Růzička M. and Šverák V. (1996). On Leray’s self-similar solutions of the Navier-Stokes equations. Acta Math. 176(2): 283–294

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. Scheffer V. (1982). Boundary regularity for the Navier-Stokes equations in a half-space. Commun. Math. Phys. 85(2): 275–299

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. Scheffer V. (1985). A solution to the Navier-Stokes inequality with an internal singularity. Commun. Math. Phys. 101(1): 47–85

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Scheffer V. (1987). Nearly one-dimensional singularities of solutions to the Navier-Stokes inequality. Commun. Math. Phys. 110(4): 525–551

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. Seregin G.A. (2002). Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary. J. Math. Fluid Mech. 4(1): 1–29

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. Seregin G.A. (2005). On Smoothness of L 3,∞-Solutions to the Navier-Stokes Equations up to Boundary. Math. Ann. 332(1): 219–238

    Article  MATH  MathSciNet  Google Scholar 

  29. Seregin, G.A., Shilkin, T.N., Solonnikov, V.A.: Boundary partial regularity for the Navier-Stokes equations. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 34, 158–190, 228; translation in J. Math. Sci. (N. Y.) 132(3), 339–358 (2006)

  30. Seregin G.A. and Sverak V. (2002). Navier-Stokes equations with lower bounds on the pressure. Ration. Mech. Anal. 163(1): 65–86

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

  33. Sohr H. (1983). Zur Regularitätstheorie der instationären Gleichungen von Navier-Stokes. Math. Z. 184(3): 359–375

    Article  MATH  MathSciNet  Google Scholar 

  34. Sohr H. (2001). A regularity class for the Navier-Stokes equations in Lorentz spaces. J. Evol. Equ. 1(4): 441–467

    Article  MATH  MathSciNet  Google Scholar 

  35. Sohr H. (1983). Zur Regularitätstheorie der instationären Gleichungen von Navier-Stokes. Math. Z. 184(3): 359–375

    Article  MATH  MathSciNet  Google Scholar 

  36. Solonnikov, V.A.: Estimates of solutions of the Stokes equations in S. L. Sobolev spaces with a mixed norm. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 288, 204–231 (2002)

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

    Article  MATH  MathSciNet  Google Scholar 

  38. Takahashi S. (1990). On interior regularity criteria for weak solutions of the Navier-Stokes equations. Manuscripta Math. 69(3): 237–254

    Article  MATH  MathSciNet  Google Scholar 

  39. Tian G. and Xin Z. (1999). Gradient estimation on Navier-Stokes equations. Comm. Anal. Geom. 7(2): 221–257

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2

    Stephen Gustafson & Tai-Peng Tsai

  2. Department of Mathematics, Sungkyunkwan University and Institute of Basic Science, Suwon, 440-746, Republic of Korea

    Kyungkeun Kang

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  1. Stephen Gustafson
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  2. Kyungkeun Kang
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  3. Tai-Peng Tsai
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Correspondence to Tai-Peng Tsai.

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Communicated by P. Constantin

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Gustafson, S., Kang, K. & Tsai, TP. Interior Regularity Criteria for Suitable Weak Solutions of the Navier-Stokes Equations. Commun. Math. Phys. 273, 161–176 (2007). https://doi.org/10.1007/s00220-007-0214-6

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