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.
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References
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)
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)
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)
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)
Constantin, P.: Nonlinear Fokker–Planck Navier–Stokes systems. Commun. Math. Sci. 3(4), 531–544 (2005)
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)
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)
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)
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)
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)
Constantin, P., Kevrekidis, I., Titi, E.S.: Remarks on a Smoluchowski equation. Discret. Contin. Dyn. Syst. 11(1), 101–112 (2004)
Constantin, P., Titi, E.S., Vukadinovic, J.: Dissipativity and Gevrey regularity of a Smoluchowski equation. Indiana Univ. Math. J. 54(4), 949–969 (2005)
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)
Doi, M., Edwards, S.: The Theory of Polymer Dynamics. Oxford University Press, Oxford (1986)
La, J.: On diffusive 2d fokker-planck-navier-stokes systems. arXiv:1804.05168 (2018)
Lin, F.: Some analytical issues for elastic complex fluids. Commun. Pure Appl. Math. 65(7), 893–919 (2012)
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)
Masmoudi, N.: Equations for polymeric materials. In: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, pp. 973–1005 (2018)
Musacchio, S., Cencini, M., Emmanuel, L., Vincenzi, D.: Enhancement of mixing by rodlike polymers. Eur. Phys. J. E 41(7), 84 (2018)
Otto, F., Tzavaras, A.E.: Continuity of velocity gradients in suspensions of rod-like molecules. Commun. Math. Phys. 277(3), 729–758 (2008)
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)
Zhang, H., Zhang, P.: On the new multiscale rodlike model of polymeric fluids. SIAM J. Math. Anal. 40(3), 1246–1271 (2008)
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.
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Communicated by Dr. Paul Newton.
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La, J. Global Well-Posedness of Strong Solutions of Doi Model with Large Viscous Stress. J Nonlinear Sci 29, 1891–1917 (2019). https://doi.org/10.1007/s00332-019-09533-8
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DOI: https://doi.org/10.1007/s00332-019-09533-8