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Existence of large-data global-in-time finite-energy weak solutions to a compressible FENE-P model

John W. Barrett

Department of Mathematics, Imperial College London, London SW7 2AZ, UK

E-mail Address: j.barrett@imperial.ac.uk

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and

Endre Süli

Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK

E-mail Address: suli@maths.ox.ac.uk

Corresponding author

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https://doi.org/10.1142/S0218202518500471Cited by:3 (Source: Crossref)

Abstract

A compressible FENE-P-type model with stress diffusion is derived from an approximate macroscopic closure of a compressible Navier–Stokes–Fokker–Planck system arising in the kinetic theory of dilute polymeric fluids, where polymer chains immersed in a barotropic, compressible, isothermal, viscous Newtonian solvent are idealised as pairs of massless beads connected with finitely extensible nonlinear elastic (FENE) springs. We develop a priori bounds for the model, including logarithmic bounds, which guarantee the non-negativity of the conformation tensor and a bound on its trace, and we prove the existence of large-data global-in-time finite-energy weak solutions in two and three space dimensions.

Communicated by N. Bellomo

Keywords:

AMSC: 35A01, 35Q35, 76A05

References

1. J. W. Barrett and S. Boyaval , Existence and approximation of a (regularized) Oldroyd-B model, Math. Models Methods Appl. Sci. 21 (2011) 1783–1837. Link, Web of Science, Google Scholar
2. J. W. Barrett and S. Boyaval , Finite element approximation of the FENE-P model, to appear in IMA J. Numer. Anal. (2018), https://academic.oup.com/imajna/advance-article/doi/10.1093/imanum/drx061/4561632. Crossref, Web of Science, Google Scholar
3. J. W. Barrett, Y. Lu and E. Süli , Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci. 15 (2017) 1265–1323. Crossref, Web of Science, Google Scholar
4. J. W. Barrett and E. Süli , Existence of global weak solutions to compressible isentropic finitely extensible nonlinear bead-spring chain models for dilute polymers, Math. Models Methods Appl. Sci. 26 (2016) 469–568. Link, Web of Science, Google Scholar
5. J. W. Barrett and E. Süli , Existence of global weak solutions to compressible isentropic finitely extensible nonlinear bead-spring chain models for dilute polymers: The two-dimensional case, J. Differential Equations 261 (2016) 592–626. Crossref, Web of Science, Google Scholar
6. R. Bird, P. Dotson and N. Johnson , Polymer solution rheology based on a finitely extensible bead-spring chain model, J. Non-Newtonian Fluid Mech. 7 (1980) 213–235. Crossref, Web of Science, Google Scholar
7. M. E. Bogovskiĭ , Solutions of some problems of vector analysis, associated with the operators div and grad, in Theory of Cubature Formulas and the Application of Functional Analysis to Problems of Mathematical Physics, Trudy Sem. S. L. Soboleva, Vol. 1980, No. 1 (Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1980), pp. 5–40, 149. Google Scholar
8. S. Dain , Generalised Korn’s inequality and conformal Killing vectors, Calc. Var. Partial Differential Equations 25 (2006) 535–540. Crossref, Web of Science, Google Scholar
9. L. C. Evans , Partial Differential Equations, 2nd edn. Graduate Studies in Mathematics, Vol. 19 (Amer. Math. Soc., 2010). Crossref, Google Scholar
10. E. Feireisl , On compactness of solutions to the compressible isentropic Navier–Stokes equations when the density is not square integrable, Comment. Math. Univ. Carolin. 42 (2001) 83–98. Google Scholar
11. E. Feireisl , Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, Vol. 26 (Oxford Univ. Press, 2004). Google Scholar
12. E. Feireisl and A. Novotný , Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics (Birkhäuser, 2009). Crossref, Google Scholar
13. E. Feireisl, A. Novotný and H. Petzeltová , On the existence of globally defined weak solutions to the Navier–Stokes equations, J. Math. Fluid Mech. 3 (2001) 358–392. Crossref, Web of Science, Google Scholar
14. G. P. Galdi , An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, Springer Monographs and Studies in Mathematics (Springer, 2011). Crossref, Google Scholar
15. D. Gilbarg and N. S. Trudinger , Elliptic Partial Differential Equations of Second Order, Classics in Mathematics (Springer, 2001). Reprint of the 1998 edition. Crossref, Google Scholar
16. P. Grisvard , Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, Vol. 24 (Pitman (Advanced Publishing Program), 1985). Google Scholar
17. D. Hu and T. Lelièvre , New entropy estimates for Oldroyd-B and related models, Commun. Math. Sci. 5 (2007) 909–916. Crossref, Web of Science, Google Scholar
18. R. Keunings , On the Peterlin approximation for finitely extensible dumbbells, J. Non-Newtonian Fluid Mech. 68 (1997) 85–100. Crossref, Web of Science, Google Scholar
19. P.-L. Lions , Mathematical Topics in Fluid Dynamics: Compressible Models, Vol. 2 (Oxford Science Publication, 1998). Google Scholar
20. J.-L. Lions and E. Magenes , Non-Homogeneous Boundary Value Problems and Applications, Vol. 1 (Springer, 1972). Translated from the French by P. Kenneth, Die Grundlehren der Mathematischen Wissenschaften, Band 181. Crossref, Google Scholar
21. N. Masmoudi , Global existence of weak solutions to macroscopic models of polymeric flows, J. Math. Pures Appl. 96 (2011) 502–520. Crossref, Web of Science, Google Scholar
22. A. Novotný and I. Straškraba , Introduction to the Mathematical Theory of Compressible Flow (Oxford Univ. Press, Oxford, 2004). Crossref, Google Scholar
23. A. Peterlin , Hydrodynamics of macromolecules in a velocity field with longitudinal gradient, Polym. Lett. 4 (1966) 287–291. Crossref, Web of Science, Google Scholar
24. J. Simon , Compact sets in the space $L^{p} (0, T; B)$ $L^{p} (0, T; B)$ , Ann. Math. Pura. Appl. 146 (1987) 65–96. Crossref, Web of Science, Google Scholar
25. W. A. Strauss , On continuity of functions with values in various Banach spaces, Pacific J. Math. 19 (1966) 543–551. Crossref, Web of Science, Google Scholar
26. P. Wapperom and M. Hulsen , Thermodynamics of viscoelastic fluids: The temperature equation, J. Non-Newtonian Fluid Mech. 42 (1998) 999–1019. Google Scholar
27. T. P. Wihler , On the Hölder continuity of matrix functions for normal matrices, JIPAM. J. Inequal. Pure Appl. Math. 10 (2009), Article 91, 5pp. Google Scholar