Partial Differential Equations
A singular asymptotic behavior of a transport equation
Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 155-159.

We consider a simple conservative transport equation where the speed is strictly decreasing. The monotonicity property of the speed rate leads to a singular asymptotic behavior and the concentration of the mass of the solution at a point. Thus, a model which contains a transport structure with monotone decay of the speed rate can be reduced by using the result of convergence to a Dirac mass. It is useful in the case where we have to simulate numerous nonlinear PDEs containing such a structure. Indeed, the concentration of the mass makes the variable in which the mass concentrate useless and thus we lose a dimension. The gain in time calculus is important when the number of equations is large.

On s'intéresse à des équations de transport dans lesquelles la vitesse de transport est strictement décroissante. Cette propriété de monotonie entraîne la concentration de la masse de la solution en un point et, par conséquent, un comportement asymptotique singulier. D'autre part, il est possible de réduire un modèle contenant une structure de transport avec la condition de monotonie (dans une direction) de la vitesse de transport. En utilisant la convergence de la solution en une masse de Dirac suivant la direction du transport, on rend cette direction inutile et on peut éliminer une variable d'espace. Cela réduit le coût numérique lorsqu'il faut simuler un certains nombres d'EDP contenant une telle structure. Le gain est d'autant plus important que le nombre d'équations est grand.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.12.010
Michel, Philippe 1

1 DMI, Institut Camille-Jordan, École centrale de Lyon, 36, avenue Guy-de-Collongue, 69134 Ecully cedex, France
@article{CRMATH_2008__346_3-4_155_0,
     author = {Michel, Philippe},
     title = {A singular asymptotic behavior of a transport equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {155--159},
     publisher = {Elsevier},
     volume = {346},
     number = {3-4},
     year = {2008},
     doi = {10.1016/j.crma.2007.12.010},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2007.12.010/}
}
TY  - JOUR
AU  - Michel, Philippe
TI  - A singular asymptotic behavior of a transport equation
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 155
EP  - 159
VL  - 346
IS  - 3-4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2007.12.010/
DO  - 10.1016/j.crma.2007.12.010
LA  - en
ID  - CRMATH_2008__346_3-4_155_0
ER  - 
%0 Journal Article
%A Michel, Philippe
%T A singular asymptotic behavior of a transport equation
%J Comptes Rendus. Mathématique
%D 2008
%P 155-159
%V 346
%N 3-4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2007.12.010/
%R 10.1016/j.crma.2007.12.010
%G en
%F CRMATH_2008__346_3-4_155_0
Michel, Philippe. A singular asymptotic behavior of a transport equation. Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 155-159. doi : 10.1016/j.crma.2007.12.010. http://www.numdam.org/articles/10.1016/j.crma.2007.12.010/

[1] Bolley, F.; Villani, C. Weighted Csiszar–Kullback–Pinsker inequalities and application to transport equation inequalities, Ann. Fac. Sci. Toulouse Ser. 6, Volume 14 (2005) no. 3, pp. 331-352

[2] Cushing, J.M. An Introduction to Structured Population Dynamics, SIAM, Philadelphia, 1998

[3] Diekmann, O.; Gyllenberg, M.; Metz, J.A.J. Steady state analysis of structured population models, Theor. Pop. Biol., Volume 63 (2003), pp. 309-338

[4] DiPerna, R.J. Measure valued solutions to conservation laws, Arch. Rational Mech. Anal., Volume 88 (1985), pp. 223-270

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

[6] DiPerna, R.J.; Lions, P.L. Solutions globales d'équations du type Vlasov–Poisson, C. R. Acad. Sci. Paris, Ser. I, Volume 307 (1988), pp. 655-658

[7] Echenim, N.; Monniaux, D.; Sorine, M.; Clement, F. Multi-scale modeling of the follicle selection process in the ovary, Math. Biosci., Volume 198 (2005) no. 1, pp. 57-79

[8] Iannelli, M. Age-structured population (Hazewinkel, M., ed.), Encyclopedia of Mathematics, supplement II, Kluwer Academics, 2000, pp. 21-23

[9] Jabin, P.E. Macroscopic limit of Vlasov type equations with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 17 (2000), pp. 651-672

[10] Jabin, P.E. Large time concentration for solutions to kinetic equations with energy dissipation, Comm. Partial Differential Equations, Volume 25 (2000), pp. 541-557

[11] Lacker, H.M.; Percus, A. How do ovarian follicles interact? A many body problem with unusual symmetry and symmetry-breaking properties, J. Stat. Phys., Volume 63 (1991), pp. 1133-1161

[12] Michel, P.; Mischler, S.; Perthame, B. General relative entropy inequality: an illustration on growth models, J. Math. Pures Appl., Volume 84 (2005) no. 9, pp. 1235-1260

[13] Mischler, S.; Perthame, B.; Ryzhik, L. Stability in a nonlinear population maturation model, Math. Models Meth. Appl. Sci., Volume 12 (2002) no. 12, pp. 1751-1772

[14] Metz, J.A.J.; Diekmann, O. The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, vol. 68, Springer-Verlag, 1986

[15] Otto, F.; Westdickenberg, M. Eulerian calculus for the contraction in the Wasserstein distance, SIAM J. Math. Anal., Volume 37 (2005), pp. 1227-1255

[16] Perthame, B. Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser, 2007

[17] Rotenberg, M. Transport theory for growing cell populations, J. Theor. Biol., Volume 103 (1983), pp. 181-199

Cited by Sources: