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Science China Mathematics

, Volume 61, Issue 4, pp 727–744 | Cite as

Existence of weak solutions for non-stationary flows of fluids with shear thinning dependent viscosities under slip boundary conditions in half space

  • Aibin Zang
Articles
  • 37 Downloads

Abstract

This paper treats the system of motion for an incompressible non-Newtonian fluids of the stress tensor described by p-potential function subject to slip boundary conditions in ℝ + 3 . Making use of the Oseentype approximation to this model and the L-truncation method, one can establish the existence theorem of weak solutions for p-potential flow with p ∈ (\(\frac{8}{5}\), 2] provided that large initial are regular enough.

Keywords

non-Newtonian fluid slip boundary conditions Oseen-type approximation weak solution 

MSC(2010)

76D05 35D05 54B15 34A34 

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Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11571279), Education Department of Jiangxi Province (Grant No. GJJ151036) and Youth Innovation Group of Applied Mathematics in Yichun University (Grant No. 2012TD006).

References

  1. 1.
    Acerbi E, Mingione G, Seregin G A. Regularity results for parabolic systems related to a class of non-Newtonian fluids. Ann Inst H Poincaré Anal Non Linéaire, 2004, 21: 25–60MathSciNetzbMATHGoogle Scholar
  2. 2.
    Adams R, Fournier J. Sobolev Spaces, 2nd ed. Singapore: Elsevier, 2009zbMATHGoogle Scholar
  3. 3.
    Amann H. Stability of the rest state of a viscous incompressible fluid. Arch Ration Mech Anal, 1994, 126: 231–242MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bae H, Choe H J. Existence and regularity of solutions of non-Newtonian flow. Quart Appl Math, 2000, 58: 379–400MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beirão da Veiga H. On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions. Comm Pure Appl Math, 2005, 58: 552–577MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Beirão da Veiga H. Navier-Stokes equations with shear thickening viscosity: Regularity up to boundary. J Math Fluid Mech, 2009, 11: 233–257MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Beirão da Veiga H. Navier-Stokes equations with shear thinning viscosity: Regularity up to boundary. J Math Fluid Mech, 2009, 11: 258–273MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Beirão da Veiga H. On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem. J Eur Math Soc (JEMS), 2009, 11: 127–167MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Beirão da Veiga H. Turbulence models, p-fluid flows, and W2;l-regularity of solutions. Comm Pure Appl Anal, 2009, 8: 769–783MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Beirão da Veiga H. On the global regularity of shear thinning flows in smooth domains. J Math Anal Appl, 2009, 349: 335–360MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Beirão da Veiga H, Crispo F. Concerning the Wk;p-inviscid limit for 3D flows under a slip boundary condition. J Math Fluid Mech, 2011, 13: 117–135MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Beirão da Veiga H, Crispo F, Grisant C R. Reducing slip boundary value problems from the half to the whole space: Applications to inviscid limits and to non-Newtonian fluids. J Math Anal Appl, 2011, 377: 216–227MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Beirão da Veiga H, Kaplický P, Růžička M. Regularity theorems, up to the boundary, for shear thickening flows. C R Math Acad Sci Paris, 2010, 348: 541–544MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Berselli L C, Diening L, Růžička M. Existence of strong solutions for incompressible fluids with shear dependent viscosities. J Math Fluid Mech, 2005, 12: 101–132MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bird R B, Armstrong R C, Hassager O. Dynamic of Polymer Liquids, 2nd ed. New York: John Wiley, 1987Google Scholar
  16. 16.
    Bothe D, Prüss J. Lp-theory for a class of non-Newtonian fluids. SIAM J Math Anal, 2007, 39: 379–421MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bulíček M, Málek J, Rajagopal K R. Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity. Indiana Univ Math J, 2007, 56: 51–85MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bulíček M, Majdoub M, Málek J. Unsteady flows of fluids with pressure dependent viscosity in unbounded domains. Nonlinear Anal Real World Appl, 2010, 11: 3968–3983MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Bulíček M, Málek J, Rajagopal K R. Mathematical analysis of unsteady flows of fluids with pressure, shear-rate, and temperature dependent material moduli that slip at solid boundaries. SIAM J Math Anal, 2009, 41: 665–707MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Crispo F. Shear thinning viscous fluids in cylindrical domain. Regularity up to the boundary. J Math Fluid Mech, 2008, 10: 311–325CrossRefzbMATHGoogle Scholar
  21. 21.
    Crispo F. Global regularity of a class of p-fluid flow in cylinders. J Math Anal Appl, 2008, 341: 559–574MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Diening L, Málek J, Steinhauer M. On the Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM Control Optim Calc Var, 2008, 14: 211–232MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Diening L, Růžička M. Strong solutions for generalized Newtonian fluids. J Math Fluid Mech, 2005, 7: 413–450MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Diening L, Růžička M, Wolf J. Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Ann Sc Norm Super Pisa Cl Sci (5), 2010, 9: 1–46MathSciNetzbMATHGoogle Scholar
  25. 25.
    Ebmeyer C. Regularity in Sobolev spaces of steady flows of fluids with shear-dependent viscosity. Math Methods Appl Sci, 2006, 29: 1687–1707MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Frehse J, Málek J, Steinhauer M. An existence result for fluids with shear dependent viscosity-steady flows. Nonlinear Anal, 1997, 30: 3041–3049MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Frehse J, Málek J, Steinhauer M. On the analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J Math Anal, 2003, 34: 1064–1083MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Galdi P, Grisanti R. Existence and regularity of steady flows for shear-thinning liquids in exterior two-dimensional domains. Arch Ration Mech Anal, 2011, 200: 533–559MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kato T, Lai C Y. Nonlinear evolution equations and the Euler flow. J Funct Anal, 1984, 56: 15–28MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ladyžhenskaya O A. New equation for description of motion of viscous incompressible fluids and solvability in the large boundary value problems for them. Proc Steklov Inst Math, 1967, 102: 95–118Google Scholar
  31. 31.
    Ladyžhenskaya O A. On some modification of the Navier-Stokes equations for large gradients of velocity. Zap Nauchn Sem S-Peterburg Otdel Mat Inst Steklov (POMI), 1968, 7: 126–154zbMATHGoogle Scholar
  32. 32.
    Ladyžhenskaya O A. The Mathematical Theory of Incompressible Flow, 2nd ed. New York: Gordon and Breach, 1969zbMATHGoogle Scholar
  33. 33.
    Lions J L. Quelques methodes de résolution des problèmes aus limites nonlinéaires. Paris: Dunod, 1969zbMATHGoogle Scholar
  34. 34.
    Málek J, Nečas J, Rokyta M, et al. Weak and Measure-Valued Solutions to Evolutionary PDEs. London: Chapman & Hall, 1996CrossRefzbMATHGoogle Scholar
  35. 35.
    Málek J, Nečas J, Růžička M. On the weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: The case p ≤ 2. Adv Difference Equ, 2001, 6: 257–302MathSciNetzbMATHGoogle Scholar
  36. 36.
    Málek J, Rajagopal K R. Mathematical issues concerning the Navier-Stokes equations and some of its generalizations. In: Evolutionary Equations. Handbook of Differential Equations, vol. 2. Amsterdam: Elsevier/North-Holland, 2005, 371–459MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Maremonti R. Some theorems of existence for solutions of the Navier-Stokes equations with slip boundary conditions in half-space. Ric Mat, 1991, 40: 81–135MathSciNetzbMATHGoogle Scholar
  38. 38.
    Naumann J, Wolf J. Interior differentiability of weak solutions to the equations of stationary motion of a class of non-Newtonian fluids. J Math Fluid Mech, 2005, 7: 298–313MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Navier C L M. Mémoire sur les lois du mouvement des fluides. Mémoires de l‘Académie Royale des Sciences de Institutde France, vol. 1. Mem Acad Sci Inst France, http://cdarve.web.cern.ch/cdarve/publications cd/navier darve.pdf, 1822Google Scholar
  40. 40.
    Pokorný M. Cauchy problem for the non-Newtonian viscous incompressible fluid. Appl Math, 1996, 41: 169–201MathSciNetzbMATHGoogle Scholar
  41. 41.
    Shinlkin T N. Regularity up to boundary of solutions to boundary-value problems of the theory of generalized Newtonian liquids. J Math Sci (NY), 1998, 92: 4386–4403MathSciNetCrossRefGoogle Scholar
  42. 42.
    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, 2007, 9: 104–138MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Xiao Y L, Xin Z P. On the vanishing viscosity limit for the Navier-Stokes equations with a slip boundary condition. Comm Pure Appl Math, 2007, 60: 1027–1055MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceYichun UniversityYichunChina

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