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Existence of Regular Time-Periodic Solutions to Shear-Thinning Fluids

  • Anna Abbatiello
  • Paolo MaremontiEmail author
Article

Abstract

In this note we investigate the existence of time-periodic solutions to the p-Navier–Stokes system in the singular case of \(p\in (1,2)\), which describes the flows of an incompressible shear-thinning fluid. In the 3D space-periodic setting and for \(p\in [\frac{5}{3},2)\) we prove the existence of a regular time-periodic solution corresponding to a time-periodic force datum that is assumed small in a suitable sense. As a particular case we obtain “regular” steady solutions.

Keywords

p-Navier-Stokes problem Time-periodic solutions Global existence 

Mathematics Subject Classification

35Q35 35B10 76A05. 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflicts of interest.

References

  1. 1.
    Abbatiello, A., Crispo, F., Maremonti, P.: Electrorheological fluids: ill posedness of uniqueness backward in time. Nonlinear Anal. 170, 47–69 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Pure and Applied Mathematics (Amsterdam), vol. 140, 2nd edn. Academic Press, Amsterdam (2003)Google Scholar
  3. 3.
    Barhoun, A., Lemlih, A.B.: A reproductive property for a class of non-Newtonian fluids. Appl. Anal. 81(1), 13–38 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Beirão da Veiga, H.: Navier–Stokes equations with shear thinning viscosity. Regularity up to the boundary. J. Math. Fluid Mech. 11(2), 258–273 (2009)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Beirão da Veiga, H.: On the global regularity of shear thinning flows in smooth domains. J. Math. Anal. Appl. 349(2), 335–360 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Berselli, L.C., Diening, L., Růžička, M.: Existence of strong solutions for incompressible fluids with shear dependent viscosities. J. Math. Fluid. Mech. 12, 101–132 (2010)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Crispo, F.: A note on the existence and uniqueness of time-periodic electro-rheological flows. Acta Appl. Math. 132, 237–250 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Crispo, F., Grisanti, C., Maremonti, P.: Singular p-laplacian parabolic system in exterior domains: higher regularity of solutions and related properties of extinction and asymptotic behavior in time. To appear in Annali Scuola Normale Superiore di Pisa.  https://doi.org/10.2422/2036-2145.201703_019
  9. 9.
    Crispo, F., Grisanti, C.R.: On the existence, uniqueness and \(C^{1,\gamma }(\overline{\Omega })\cap W^{2,2}(\Omega )\) regularity for a class of shear-thinning fluids. J. Math. Fluid Mech. 10, 455–487 (2008)Google Scholar
  10. 10.
    Crispo, F., Grisanti, C.R.: On the \(C^{1,\gamma }(\overline{\Omega })\cap W^{2,2}(\Omega )\) regularity for a class of electro-rheological fluids. J. Math. Anal. Appl. 356(1), 119–132 (2009)Google Scholar
  11. 11.
    Crispo, F., Maremonti, P.: Higher regularity of solutions to the singular \(p\)-Laplacian parabolic system. Adv. Differ. Equ. 18(9–10), 849–894 (2013)Google Scholar
  12. 12.
    Di Benedetto, E.: Degenerate Parabolic Equations. Springer, Berlin (1993)CrossRefGoogle Scholar
  13. 13.
    Diening, L., Ebmeyer, C., Růžička, M.: Optimal convergence for the implicit space-time discretization of parabolic systems with \(p\)-structure. SIAM J. Numer. Anal. 45(2), 457–472 (2007)Google Scholar
  14. 14.
    Diening, L., Růžička, M.: Strong solutions for generalized Newtonian fluids. J. Math. Fluid Mech. 7, 413–450 (2005)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Galdi, G.P.: Mathematical problems in classical and non-Newtonian fluid mechanics. In: Hemodynamical Flows, Oberwolfach Semin., vol. 37 (pp. 121–273). Birkhäuser, Basel (2008)Google Scholar
  16. 16.
    Lions, J.L.: Sur certaines équations paraboliques non linéaires. Bull. Soc. Math. France 93, 155–175 (1965)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Málek, J., Nečas, J., Rokyta, M., Růžička, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs, Applied Mathematics and Mathematical Computation, vol. 13. Chapman & Hall, London (1996)zbMATHGoogle Scholar
  18. 18.
    Málek, J., Rajagopal, K.R., Růžička, M.: Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity. Math. Models Methods Appl. Sci. 5(6), 789–812 (1995)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Maremonti, P.: Existence and stability of time-periodic solutions to the Navier–Stokes equations in the whole space. Nonlinearity 4(2), 503–529 (1991)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Prodi, G.: Qualche risultato riguardo alle equazioni di Navier–Stokes nel caso bidimensionale. Rend. Sem. Mat. Univ. Padova 30, 1–15 (1960)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Prouse, G.: Soluzioni periodiche dell’equazione di Navier–Stokes. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 35, 443–447 (1963)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Serrin, J.: A note on the existence of periodic solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 3, 120–122 (1959)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaUniversità degli Studi della Campania “Luigi Vanvitelli”CasertaItaly

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