Regularity for Ostwald-de Waele type shear thickening fluids

  • Hyeong-Ohk BaeEmail author
  • Kyungkeun Kang
  • Jihoon Lee
  • Jörg Wolf


We obtain local in time existence of strong solution for non-Newtonian fluid with shear thickening viscosity. We also obtain the Hausdorff dimension of time singular set, and a Serrin type regularity criterion.

Mathematics Subject Classification (2010)

76D03 76A05 35Q35 


Shear thickening fluid Strong solution Serrin criterion Hausdorff dimension Singularity Regularity Non-Newtonian fluid 


  1. 1.
    Bae, H.-O., Choe, H.J.: L -bound of weak solutions to Navier-Stokes equations. In: Proceedings of the Korea-Japan Partial Differential Equations Conference (Taejon, 1996). Lecture Notes Ser. 39. Seoul Nat. Univ., Seoul, p. 13 (1997)Google Scholar
  2. 2.
    Bae H.-O., Choe H.J.: A regularity criterion for the Navier-Stokes equations. Comm. Partial Differ. Equ. 32, 1173–1187 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bae H.-O., Choe H.J.: Existence of weak solutions to a class of non-newtonian flows. Houst. J. Math. 26, 387–408 (2000)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bae, H.-O., Choe, H.J., Kim, D.W.: Regularity and singularity of weak solutions to Ostwald-de Waele flows. International Conference on Differential Equations and Related Topics (Pusan, 1999). J. Korean Math. Soc. 37(6), 957–975 (2000)Google Scholar
  5. 5.
    da Veiga H.B.: On the smoothness of a class of weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 2, 315–323 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Berselli L.C., Diening L., Ruzicka M.: Existence of strong solutions for incompressible fluids with shear dependent viscosities. J. Math. Fluid Mech. 12, 101–132 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Chae D., Choe H.J.: Regularity of solutions to the Navier-Stokes equation. Electron. J. Differ. Equ. 5, 1–7 (1999)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Chae, D., Lee, J.: Regularity criterion in terms of pressure for the Navier-Stokes equations. Nonlinear Anal. 46(5), Ser. A: Theory Methods, 727–735 (2001)Google Scholar
  9. 9.
    Diening L., Ruzicka M.: Strong solutions for generalized Newtonian fluids. J. Math. Fluid Mech. 7, 413–450 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Diening L., Ruzicka M., Wolf J.: Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9(1), 1–46 (2010)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Fabes E.B., Jones B.F., Riviere N.M.: The initial value problem for the Navier-Stokes equations with data in L p. Arch. Ration. Mech. Anal. 45, 222–240 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Gala S.: Regularity criterion on weak solutions to the Navier-Stokes equations. J. Korean Math. Soc. 45, 537–558 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kukavica I., Ziane M.: One component regularity for the Navier-Stokes equations. Nonlinearity 19(2), 453–469 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow, 2nd edn. Gordon and Breach, New York (1969)Google Scholar
  15. 15.
    Málek, J., Nečas, J., Rokyta, M., Ružička, M.: Weak and Measure-valued Solutions to Evolutionary PDEs. Chapman & Hall (1996)Google Scholar
  16. 16.
    Malek, J., Necas, J., Ruzicka, M.: On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case p ≥ 2. Adv. Differ. Equ. 6(3), 257–302 (2001)Google Scholar
  17. 17.
    Neustupa J., Novotny A., Penel P.: An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity. Top. Math. Fluid Mech. Quad. Mat. 10, 163–183 (2002)MathSciNetGoogle Scholar
  18. 18.
    Pokorny M.: Cauchy problem for the non-Newtonian viscous incompressible fluid. Appl. Math. 41(3), 169–201 (1996)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Prodi G.: Un teorema di unicit per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48(4), 173–182 (1959) (Italian)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Serrin J.: On the interior regularity of weak solutions of the Navier Stokes equations. Arch. Ration. Mech. Anal. 9, 187–195 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Wolf J.: Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity. J. Math. Fluid Mech. 9(1), 104–138 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Zhou Y.: A new regularity criterion for the Navier-Stokes equations in terms of the gradient of one velocity component. Methods Appl. Anal. 9(4), 563–578 (2002)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Hyeong-Ohk Bae
    • 1
    Email author
  • Kyungkeun Kang
    • 2
  • Jihoon Lee
    • 3
  • Jörg Wolf
    • 4
  1. 1.Department of Financial EngineeringAjou UniversitySuwonRepublic of Korea
  2. 2.Department of MathematicsYonsei UniversitySeoulKorea
  3. 3.Department of MathematicsChung-Ang UniversitySeoulKorea
  4. 4.Department of MathematicsHumboldt-University of BerlinBerlinGermany

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