no. 5
Existence and uniqueness of a density probability solution for the stationary Doi–Edwards equation
Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 5, pp. 1353-1373.

We prove the existence, uniqueness and non-negativity of solutions for a nonlinear stationary Doi–Edwards equation. The existence is proved by a perturbation argument. We get the uniqueness and the non-negativity by showing the convergence in time of the solution of the evolutionary Doi–Edwards equation towards any stationary solution.

DOI : 10.1016/j.anihpc.2015.05.003
Mots clés : Polymeric fluids, Doi–Edwards equation, Stationary equation, Well-posedness 
@article{AIHPC_2016__33_5_1353_0,
     author = {Ciuperca, Ionel Sorin and Heibig, Arnaud},
     title = {Existence and uniqueness of a density probability solution for the stationary {Doi{\textendash}Edwards} equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1353--1373},
     publisher = {Elsevier},
     volume = {33},
     number = {5},
     year = {2016},
     doi = {10.1016/j.anihpc.2015.05.003},
     zbl = {1356.35254},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2015.05.003/}
}
TY  - JOUR
AU  - Ciuperca, Ionel Sorin
AU  - Heibig, Arnaud
TI  - Existence and uniqueness of a density probability solution for the stationary Doi–Edwards equation
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2016
SP  - 1353
EP  - 1373
VL  - 33
IS  - 5
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2015.05.003/
DO  - 10.1016/j.anihpc.2015.05.003
LA  - en
ID  - AIHPC_2016__33_5_1353_0
ER  - 
%0 Journal Article
%A Ciuperca, Ionel Sorin
%A Heibig, Arnaud
%T Existence and uniqueness of a density probability solution for the stationary Doi–Edwards equation
%J Annales de l'I.H.P. Analyse non linéaire
%D 2016
%P 1353-1373
%V 33
%N 5
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2015.05.003/
%R 10.1016/j.anihpc.2015.05.003
%G en
%F AIHPC_2016__33_5_1353_0
Ciuperca, Ionel Sorin; Heibig, Arnaud. Existence and uniqueness of a density probability solution for the stationary Doi–Edwards equation. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 5, pp. 1353-1373. doi : 10.1016/j.anihpc.2015.05.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2015.05.003/

[1] Abraham, R.; Marsden, J.E. Foundation of Mechanics, Addison–Wesley Publishing Company, Inc., 1987

[2] Chupin, L. Fokker–Planck equation in bounded domain, Ann. Inst. Fourier, Volume 60 (2010) no. 1, pp. 217–255 | DOI | Numdam | Zbl

[3] Chupin, L. The FENE model for viscoelastic thin film flows, Methods Appl. Anal., Volume 16 (2009) no. 2, pp. 217–261 | DOI | Zbl

[4] Ciuperca, I.S.; Palade, L.I. The steady state configurational distribution diffusion equation of the standard FENE dumbbell polymer model: existence and uniqueness of solutions for arbitrary velocity gradients, Math. Models Methods Appl. Sci., Volume 19 (2009), pp. 2039–2064 | DOI | Zbl

[5] Ciuperca, I.S.; Heibig, A.; Palade, L.I. Existence and uniqueness results for the Doi–Edwards polymer melt model: the case of the (full) nonlinear configurational probability density equation, Nonlinearity, Volume 25 (2012) no. 4, pp. 991–1009 | DOI | Zbl

[6] Constantin, P.; Masmoudi, N. Global well-posedness for a Smoluchowski equation coupled with Navier–Stokes equations in 2D, Commun. Math. Phys., Volume 278 (2008), pp. 179–191 | DOI | Zbl

[7] Doi, M.; Edwards, S.F. Dynamics of concentrated polymer systems, Part 1. Brownian motion in the equilibrium state, J. Chem. Soc. Faraday Trans. II, Volume 74 (1978), pp. 789–1801 | DOI

[8] Doi, M.; Edwards, S.F. Dynamics of concentrated polymer systems, Part 3. The constitutive equation, J. Chem. Soc. Faraday Trans. II, Volume 74 (1978), pp. 1818–1832

[9] Doi, M.; Edwards, S.F. The Theory of Polymer Dynamics, Oxford University Press, Oxford, 1989

[10] de Gennes, P.G. Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, London, 1979

[11] DiPerna, R.J.; Lions, P.L. Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., Volume 98 (1989), pp. 511–547 | DOI | Zbl

[12] Henrot, A.; Pierre, M. Variation et optimisation de forme, une analyse géométrique, Springer, Berlin, 2005 | Zbl

[13] Jourdain, B.; Lelièvre, T.; Le Bris, C. Existence of solution for a micro–macro model of polymeric fluid: the FENE model, J. Funct. Anal., Volume 209 (2004), pp. 162–193 | DOI | Zbl

[14] Jourdain, B.; Lelièvre, T.; Le Bris, C.; Otto, F. Long time asymptotics of a multiscale model for polymeric fluid flows, Arch. Ration. Mech. Anal., Volume 181 (2006), pp. 97–148 | DOI | Zbl

[15] Masmoudi, N. Well-posedness for the FENE dumbbell model of polymeric flows, Commun. Pure Appl. Math., Volume 61 (2008) no. 12, pp. 1685–1714 | DOI | Zbl

[16] Otto, F.; Tzavaras, A. Continuity of velocity gradients in suspensions of rod-like molecules, Commun. Math. Phys., Volume 277 (2008), pp. 729–758 | DOI | Zbl

[17] Palierne, J.F. Rheothermodynamics of the Doi–Edwards reptation model, Phys. Rev. Lett., Volume 93 (2004) (136001-1–136001-4) | DOI

Cité par Sources :