Advertisement

Global Well-Posedness of Strong Solutions of Doi Model with Large Viscous Stress

  • Joonhyun LaEmail author
Article
  • 15 Downloads

Abstract

We study models of dilute rigid rod-like polymer solutions. We establish the global well-posedness of the Doi model for large data and for arbitrarily large viscous stress parameter. The main ingredient in the proof is the fact that the viscous stress adds dissipation to high derivatives of velocity.

Keywords

Complex fluids Polymeric solutions Doi model 

Mathematics Subject Classification

35 76 

Notes

Acknowledgements

The author deeply appreciates the helpful support of Professor Peter Constantin. Professor Constantin encouraged the development of the paper and gave a lot of helpful comments. Doctor Dario Vincenzi is the one who introduced the author to work on (4), and the author also thanks to Doctor Vincenzi. Research of the author was partially supported by Samsung scholarship. Also the author thanks referees for their helpful comments.

References

  1. Bae, H., Trivisa, K.: On the Doi model for the suspensions of rod-like molecules in compressible fluids. Math. Models Methods Appl. Sci. 22(10), 1250027 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bae, H., Trivisa, K.: On the Doi model for the suspensions of rod-like molecules: global-in-time existence. Commun. Math. Sci. 11(3), 831–850 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bae, H., Trivisa, K.: On the Doi model for the suspensions of rod-like molecules in compressible fluids. In: Hyperbolic Problems: Theory, Numerics, Applications, Volume 8 of AIMS Series of Applied Mathematics, pp. 285–292. American Institute of Mathematical Sciences (AIMS), Springfield (2014)Google Scholar
  4. Bogachev, V.I., Krylov, N.V., Röckner, M., Shaposhnikov, S.V.: Fokker–Planck–Kolmogorov Equations, Volume 207 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2015)Google Scholar
  5. Constantin, P.: Nonlinear Fokker–Planck Navier–Stokes systems. Commun. Math. Sci. 3(4), 531–544 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Constantin, P.: Smoluchowski Navier–Stokes systems. In: Stochastic Analysis and Partial Differential Equations. Volume 429 of Contemporary Mathematics, pp. 85–109. American Mathematical Society, Providence (2007)Google Scholar
  7. Constantin, P.: Remarks on complex fluid models. In: Mathematical Aspects of Fluid Mechanics. Volume 402 of London Mathematical Society, Lecture Note Series, pp. 70–87. Cambridge University Press, Cambridge (2012)Google Scholar
  8. Constantin, P., Masmoudi, N.: Global well-posedness for a Smoluchowski equation coupled with Navier–Stokes equations in 2D. Commun. Math. Phys. 278(1), 179–191 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Constantin, P., Seregin, G.: Global regularity of solutions of coupled Navier–Stokes equation and nonlinear Fokker–Planck equations. Discret. Contin. Dyn. Syst. 26(4), 1185–1196 (2010a)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Constantin, P., Seregin, G.: Hölder continuity of solutions of 2D Navier-Stokes equations with singular forcing. In: Nonlinear Partial Differential Equations and Related Topics. Volume 229 of American Mathematical Society Translations: Series 2, pp. 87–95. American Mathematical Society, Providence (2010b)Google Scholar
  11. Constantin, P., Kevrekidis, I., Titi, E.S.: Remarks on a Smoluchowski equation. Discret. Contin. Dyn. Syst. 11(1), 101–112 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Constantin, P., Titi, E.S., Vukadinovic, J.: Dissipativity and Gevrey regularity of a Smoluchowski equation. Indiana Univ. Math. J. 54(4), 949–969 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Constantin, P., Fefferman, C., Titi, E.S., Zarnescu, A.: Regularity of coupled two-dimensional nonlinear Fokker–Planck and Navier–Stokes systems. Commun. Math. Phys. 270(3), 789–811 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Doi, M., Edwards, S.: The Theory of Polymer Dynamics. Oxford University Press, Oxford (1986)Google Scholar
  15. La, J.: On diffusive 2d fokker-planck-navier-stokes systems. arXiv:1804.05168 (2018)
  16. Lin, F.: Some analytical issues for elastic complex fluids. Commun. Pure Appl. Math. 65(7), 893–919 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Lions, P.-L., Masmoudi, N.: Global existence of weak solutions to some micro-macro models. C. R. Math. Acad. Sci. Paris 345(1), 15–20 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Masmoudi, N.: Equations for polymeric materials. In: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, pp. 973–1005 (2018)Google Scholar
  19. Musacchio, S., Cencini, M., Emmanuel, L., Vincenzi, D.: Enhancement of mixing by rodlike polymers. Eur. Phys. J. E 41(7), 84 (2018)CrossRefGoogle Scholar
  20. Otto, F., Tzavaras, A.E.: Continuity of velocity gradients in suspensions of rod-like molecules. Commun. Math. Phys. 277(3), 729–758 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Wang, W., Zhang, P., Zhang, Z.: The small Deborah number limit of the Doi–Onsager equation to the Ericksen–Leslie equation. Commun. Pure Appl. Math. 68(8), 1326–1398 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Zhang, H., Zhang, P.: On the new multiscale rodlike model of polymeric fluids. SIAM J. Math. Anal. 40(3), 1246–1271 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations