Asymptotics of the KPP minimal speed within large drift

Mohammad El Smaily; Stéphane Kirsch

doi:10.1016/j.crma.2010.07.007

Mathematical Analysis/Partial Differential Equations

Asymptotics of the KPP minimal speed within large drift

Presented by: Jean-Michel Bony

Mohammad El Smaily ¹; Stéphane Kirsch ¹

¹ Department of Mathematics, University of British Columbia & Pacific Institute for the Mathematical Sciences, 1984 Mathematics Road, V6T 1Z2, Vancouver, BC, Canada

Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 857-861.

Abstract
Résumé

This Note is concerned with the asymptotic behavior of the minimal KPP speed of propagation for reaction–advection–diffusion equations with a large drift Mq (where q is the advection). We first give the limit of the speed as $M \to + \infty$ in any space dimension N. Then, we give the necessary and sufficient condition that the advection field should satisfy so that the speed acts as $O (M)$ as $M \to + \infty$ .

Dans cette Note on étudie le comportement asymptotique de la vitesse minimale de propagation des fronts progressifs pulsatoires satisfaisant une équation de réaction–advection–diffusion dans le cas d'une grande advection Mq (où q est l'advection). On donne la valeur limite de la vitesse lorsque $M \to + \infty$ dans un espace de dimension N quelconque. Pour le cas $N = 2$ on donne une condition nécessaire et suffisante pour que la vitesse se comporte comme $O (M)$ pour $M \to + \infty$ .

Received: 2010-04-20
Accepted: 2010-07-08
Published online: 2010-08-01

DOI: 10.1016/j.crma.2010.07.007

Author's affiliations:

Mohammad El Smaily ¹; Stéphane Kirsch ¹

¹ Department of Mathematics, University of British Columbia & Pacific Institute for the Mathematical Sciences, 1984 Mathematics Road, V6T 1Z2, Vancouver, BC, Canada

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     author = {Mohammad El Smaily and St\'ephane Kirsch},
     title = {Asymptotics of the {KPP} minimal speed within large drift},
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     year = {2010},
     doi = {10.1016/j.crma.2010.07.007},
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Mohammad El Smaily; Stéphane Kirsch. Asymptotics of the KPP minimal speed within large drift. Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 857-861. doi : 10.1016/j.crma.2010.07.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.007/

References
Cited by

[1] H. Berestycki; F. Hamel Front propagation in periodic excitable media, Commun. Pure Appl. Math., Volume 55 (2002), pp. 949-1032

[2] H. Berestycki; F. Hamel; N. Nadirashvili The principal eigenvalue of elliptic operators with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys., Volume 253 (2005), pp. 451-480

[3] H. Berestycki; F. Hamel; N. Nadirashvili The speed of propagation for KPP type problems. I – Periodic framework, J. Eur. Math. Soc., Volume 7 (2005), pp. 173-213

[4] M. El Smaily Pulsating travelling fronts: Asymptotics and homogenization regimes, European J. Appl. Math., Volume 19 (2008), pp. 393-434

[5] M. El Smaily; F. Hamel; L. Roques Homogenization and influence of fragmentation in a biological invasion model, Discrete Contin. Dyn. Syst. A, Volume 25 (2009), pp. 321-342

[6] M. El Smaily Min–Max formulae for the speeds of pulsating travelling fronts in periodic excitable media, Ann. Mat. Pura Appl., Volume 189 (2010), pp. 47-66

[7] M. El Smaily; S. Kirsch The speed of propagation for KPP reaction–diffusion equations within large drift, 2009 (preprint) | arXiv

[8] A. N Kolmogorov; I. G Petrovsky; N. S Piskunov Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. d'Etat de Moscou (Bjul. Moskowskogo Gos. Univ.) Sér. A, Volume 1 (1937), pp. 1-26

[9] A. Zlatoš Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows, Arch. Ration. Mech. Anal., Volume 195 (2010), pp. 441-453

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