Mathematical Problems in Mechanics
A kinetic approximation of Hele–Shaw flow
Comptes Rendus. Mathématique, Volume 338 (2004) no. 2, pp. 177-182.

In this Note we consider a fourth order degenerate parabolic equation modeling the evolution of the interface of a spreading droplet. The equation is approximated trough a collisional kinetic equation. This permits to derive numerical approximations that preserves positivity of the solution and the main relevant physical properties. A Monte Carlo application is also shown.

Dans cette Note, nous considérons une équation dégénérée du quatrième ordre modélisant l'évolution de l'interface d'une goutte. L'équation est approchée par une équation collisionnelle cinétique. Cela permet de construire des approximations numériques qui préservent la positivité de la solution et ses principales propriétés physiques. Un exemple « Monte-Carlo  » est aussi présenté.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.11.006
Pareschi, Lorenzo 1; Russo, Giovanni 2; Toscani, Giuseppe 3

1 Department of Mathematics, University of Ferrara, via Machiavelli 35, 35100 Ferrara, Italy
2 Department of Mathematics and Computer Science, University of Catania, viale Andrea Doria 6, 95125 Catania, Italy
3 Department of Mathematics, University of Pavia, via Ferrata 1, 27100 Pavia, Italy
@article{CRMATH_2004__338_2_177_0,
     author = {Pareschi, Lorenzo and Russo, Giovanni and Toscani, Giuseppe},
     title = {A kinetic approximation of {Hele{\textendash}Shaw} flow},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {177--182},
     publisher = {Elsevier},
     volume = {338},
     number = {2},
     year = {2004},
     doi = {10.1016/j.crma.2003.11.006},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2003.11.006/}
}
TY  - JOUR
AU  - Pareschi, Lorenzo
AU  - Russo, Giovanni
AU  - Toscani, Giuseppe
TI  - A kinetic approximation of Hele–Shaw flow
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 177
EP  - 182
VL  - 338
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2003.11.006/
DO  - 10.1016/j.crma.2003.11.006
LA  - en
ID  - CRMATH_2004__338_2_177_0
ER  - 
%0 Journal Article
%A Pareschi, Lorenzo
%A Russo, Giovanni
%A Toscani, Giuseppe
%T A kinetic approximation of Hele–Shaw flow
%J Comptes Rendus. Mathématique
%D 2004
%P 177-182
%V 338
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2003.11.006/
%R 10.1016/j.crma.2003.11.006
%G en
%F CRMATH_2004__338_2_177_0
Pareschi, Lorenzo; Russo, Giovanni; Toscani, Giuseppe. A kinetic approximation of Hele–Shaw flow. Comptes Rendus. Mathématique, Volume 338 (2004) no. 2, pp. 177-182. doi : 10.1016/j.crma.2003.11.006. http://www.numdam.org/articles/10.1016/j.crma.2003.11.006/

[1] Bernis, F. Finite speed of propagation and continuity of the interface for thin viscous flows, Adv. Differential Equations, Volume 3 (1996), pp. 337-368

[2] Bernis, F.; Peletier, L.A.; Williams, S.M. Source type solutions of a fourth order nonlinear degenerate parabolic equation, Nonlinear Anal., Volume 18 (1992), pp. 217-234

[3] Bertozzi, A.L. The mathematics of moving contact lines in thin liquid films, Notices Amer. Math. Soc., Volume 45 (1998), pp. 689-697

[4] Bertozzi, A.L.; Pugh, M. The lubrication approximation for thin viscous films: regularity and long-time behavior of weak solutions, Comm. Pure Appl. Math., Volume XLIX (1996), pp. 85-123

[5] Bobylev, A.V.; Nanbu, K. Theory of collision algorithms for gases and plasmas based on the Boltzmann equation and the Landau–Fokker–Planck equation, Phys. Rev. E, Volume 61 (2000), pp. 4576-4586

[6] Cercignani, C.; Illner, R.; Pulvirenti, M. The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1995

[7] Degond, P.; Lucquin-Desreux, B. The Fokker–Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Math. Models Methods Appl. Sci., Volume 2 (1992), pp. 167-182

[8] Grün, G.; Rumpf, M. Nonnegativity preserving convergent schemes for the thin films equation, Numer. Math., Volume 87 (2000), pp. 113-152

[9] Myers, T.G. Thin films with high surface tension, SIAM Rev., Volume 40 (1998), pp. 441-462

[10] Pareschi, L.; Russo, G. An introduction to Monte Carlo methods for the Boltzmann equation, ESAIM: Proceedings, Volume 10 (2001), pp. 35-75

[11] Toscani, G. One-dimensional kinetic models of granular flows, RAIRO Modél Math. Anal. Numér., Volume 34 (2000), pp. 1277-1292

[12] Toscani, G.; Villani, C. Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Statist. Phys., Volume 94 (1999), pp. 619-637

[13] Villani, C. On a new class of weak solutions for the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., Volume 143 (1998), pp. 273-307

[14] Zhornitskaya, L.; Bertozzi, A.L. Positivity preserving numerical schemes for lubrication-type equations, SIAM J. Numer. Anal., Volume 37 (2000), pp. 523-555

Cited by Sources: