Analysis of polymeric flow models and related compactness theorems in weighted spaces

SIAM Journal on Mathematical Analysis

Volumn 45, Issue 3, 2013, Pages 1179-1215

  Chen, Xiuqing ^a,b   Liu, Jian Guo ^b  

^a BEIJING UNIVERSITY OF POSTS AND TELECOMMUNICATIONS   (China)

^b DUKE UNIVERSITY   (United States)

Author keywords

Compact embedding theorem; FENE; Fokker Planck equation; Hardy type inequality; Hookean; Logarithmic Sobolev inequality; Navier Stokes equation; Polymer

Indexed keywords

COMPACT EMBEDDING; FENE; HARDY-TYPE INEQUALITY; HOOKEAN; LOGARITHMIC SOBOLEV INEQUALITY;

NAVIER STOKES EQUATIONS; TIMING JITTER; VISCOUS FLOW;

POLYMERS;

EID: 84884765570     PISSN: 00361410     EISSN: 10957111     Source Type: Journal    
DOI: 10.1137/120887850     Document Type: Article

References (31)

1. F. Antoci, Some necessary and some sufficient conditions for the compactness of the embedding of weighted Sobolev spaces, Ricerche Mat., 52 (2003), pp. 55-71.
2. A. Arnold, J. A. Carrillo, and C. Manzini, Refined long-time asymptotics for some polymeric fluid flow models, Comm. Math. Sci., 8 (2010), pp. 763-782.
3. A. Arnold, P. Markowich, G. Toscani, and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), pp. 43-100.
4. H. Bae and K. Trivisa, On the Doi model for the suspensions of rod-like molecules: Globalin-time existence, Comm. Math. Sci., 2012, to appear.
5. J. W. Barrett and E. Süli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: Finitely extensible nonlinear bead-spring chains, Math. Models Methods Appl. Sci., 21 (2011), pp. 1211-1289.
6. J. W. Barrett and E. Süli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers II: Hookean-type bead-spring chains, Math. Models Methods Appl. Sci., 22 (2012), 1150024.
7. J. W. Barrett and E. Süli, Reflections on Dubinski's nonlinear compact embedding theorem, Publ. Inst. Math. (Beograd) (N.S.), 91 (2012), pp. 95-110.
8. A. Bonito, P. Clément, and M. Picasso, Mathematical analysis of a simplified Hookean dumbbells model arising from viscoelastic flows, J. Evol. Equations, 6 (2006), pp. 381-398.
9. C. L. Bris and T. Lelièvre, Multiscale modelling of complex fluids: A mathematical initiation, in Multiscale Modeling and Simulation in Science, Lect. Notes Comput. Sci. Eng. 66, Springer, Berlin, 2009, pp. 49-137.
10. X. Q. Chen, A. Jüngel, and J.-G. Liu, Generalized Aubin-Lions Lemmas, manuscript, 2012.
11. P. Constantin, Nonlinear Fokker-Planck Navier-Stokes systems, Comm. Math. Sci., 3 (2005), pp. 531-544.
12. P. Constantin, C. Fefferman, E. S. Titi, and A. Zarnescu, Regularity of coupled twodimensional nonlinear Fokker-Planck and Navier-Stokes systems, Comm. Math. Phys., 270 (2007), pp. 789-811.
13. P. Constantin and N. Masmoudi, Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D, Comm. Math. Phys., 278 (2008), pp. 179-191.
14. P. Constantin and G. Seregin, Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker-Planck equations, Discrete Contin. Dynam. Systems, 26 (2010), pp. 1185-1196.
15. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, Oxford, 1986.
16. M. Dreher and A. Jüngel, Compact families of piecewise constant functions in Lp(0, T;B), Nonlinear Anal., 75 (2012), pp. 3072-3077.
17. J. A. Dubinskii, Weak convergence for nonlinear elliptic and parabolic equations, Mat. Sb., 67 (1965), pp. 609-642.
18. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.
19. J. G. Hooton, Compact Sobolev imbeddings on finite measure spaces, J. Math. Anal. Appl., 83 (1981), pp. 570-581.
20. M. Ledoux, The Concentration of Measure Phenomenon, AMS, Providence, RI, 2001.
21. B. Jourdain, C. L. Bris, T. Lelièvre, and F. Otto, Long-time asymptotics of a multiscale model for polymeric fluid flows, Arch. Ration. Mech. Anal., 181 (2006), pp. 97-148.
22. J. Kacur, Method of Rothe in Evolutions, Teubner Texte zur Mathematik 80, B.G. Teubner, Leipzig, 1985.
23. T. Li and P. Zhang, Mathematical analysis of multi-scale models of complex fluids, Comm. Math. Sci., 5 (2007), pp. 1-51.
24. P. L. Lions and N. Masmoudi, Global solutions of weak solutions to some micromacro models, C. R. Acad. Sci. Paris Sér. I Math., 345 (2007), pp. 15-20.
25. C. Liu and H. Liu, Boundary conditions for the microscopic FENE models, SIAM J. Appl. Math., 68 (2008), pp. 1304-1315.
26. N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows, Invent. Math., 191 (2013), pp. 427-500.
27. B. Opic, Necessary and sufficient conditions for imbeddings in weighted Sobolev spaces, Casopis pest. Mat., 114 (1989), pp. 343-355.
28. E. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben, Math. Ann., 102 (1930), pp. 650-670.
29. J. Simon, Compact sets in the space Lp(0, T;B), Ann. Mat. Pura Appl. (4), 146 (1987), pp. 65-96.
30. Y. Z. Sun and Z. F. Zhang, Global well-posedness for the 2D micro-macro models in the bounded domain, Comm. Math. Phys., 303 (2011), pp. 361-383.
31. L. Zhang, H. Zhang, and P. Zhang, Global existence of weak solutions to the regularized Hookean dumbbell model, Comm. Math. Sci., 6 (2008), pp. 85-124.