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Essential m-dissipativity of Kolmogorov operators for the 2D-stochastic shear thickening fluids

  • Desheng YangEmail author
Article
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Abstract

We study a class of viscous, incompressible non-Newtonian fluids in two space dimensions with periodic boundary conditions and an additive Gaussian noise. The nonlinear elliptic operator related to the stress tensor possesses p-structure. When the fluid is shear thickening, we prove that the associated Kolmogorov operator is essentially m-dissipative in a space with respect to an invariant measure. In addition, if the viscosity constant is sufficiently large, we show that the invariant measure is exponential mixing so that it is also unique.

Keywords

Non-Newtonian fluid Kolmogorov operator Dissipativity Invariant measure 

Mathematics Subject Classification

Primary 60H15 76A05 Secondary 47H06 37L40 

Notes

References

  1. 1.
    Barbu, V., Da Prato, G., Debussche, A.: Essential m-dissipativity of Kolmogorov operators corresponding to periodic 2D-Navier Stokes equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 15, 29–38 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Barbu, V., Da Prato, G., Debussche, A.: The Kolmogorov equation associated to the stochastic Navier–Stokes equations in 2D. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7(2), 163–182 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barbu, V., Da Prato, G.: The Kolmogorov equation for a 2D-Navier–Stokes stochastic flow in a channel. Nonlinear Anal. 69(3), 940–949 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bellout, H., Bloom, F.: Incompressible Bipolar and Non-Newtonian Viscous Fluid Flow. Advances in Mathematical Fluid Mechanics. Springer, Berlin (2014)CrossRefGoogle Scholar
  5. 5.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)CrossRefGoogle Scholar
  6. 6.
    Da Prato, G., Zabczyk, J.: Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society Lecture Note Series, vol. 293. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  7. 7.
    Flandoli, F., Maslowski, B.: Ergodicity of the 2-D Navier–Stokes equation under random perturbations. Commun. Math. Phys. 171, 119–141 (1995)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Liu, W., Röckner, M.: SPDE in Hilbert space with locally monotone coefficients. J. Funct. Anal. 259(11), 2902–2922 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Málek, J., Nec̆as, J., Rokyta, M., Ruzicka, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs. Applied Mathematics and Mathematical Computation, vol. 13. Chapman and Hall, London (1996)Google Scholar
  10. 10.
    Priola, E.: On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Stud. Math. 136(3), 271–295 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Sauer, M.: Kolmogorov equations for randomly perturbed generalized Newtonian fluids. Math. Nachr. 287(17–18), 2102–2115 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Stannat, W.: A new a priori estimate for the Kolmogorov operator of a 2D-stochastic Navier–Stokes equation. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10(4), 483–497 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Stannat, W.: Lp-uniqueness of Kolmogorov operators associated with 2D-stochastic Navier–Stokes–Coriolis equations. Math. Nachr. 284(17–18), 2287–2296 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis. SIAM, Philadelphia (1983)zbMATHGoogle Scholar
  15. 15.
    Terasawa, Y., Yoshida, N.: Stochastic power law fluids: existence and uniqueness of weak solutions. Ann. Appl. Probab. 21(5), 1827–1859 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Yoshida, N.: Stochastic shear thickening fluids: strong convergence of the Galerkin approximation and the energy equality. Ann. Appl. Probab. 22(3), 1215–1242 (2012)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaChina

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