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Mathematics Colloquium

Friday, 25 April, 2014

SPEAKER: Prof. El Smailey, Univ. of Toronto

TITLE: What kind of incompressible flows speed-up population spreading?

ABSTRACT: It is well known that reaction-advection-diffusion (RDA) equations are good models to describe population dynamics in a heterogeneous environment. Moreover, these equations model evolution of quantities such as densities of chemicals or animal species, or temperatures of combusting media which are subject to diffusion in an environment as well as a reactive process (chemical reaction, birth and death, combustion) and/or transport by a flow. When the reaction term (nonlinearity) is of the Kolomogrov-Piskunov-Petrovski (KPP for short), the RDA model admits a minimal speed c* beyond which there exists pulsating traveling wave solutions with speed c greater  or equal to c*. Unfortunately, the minimal KPP speed c* doas not have a simple formula. Instead there is a variational formalu for this threshold c* which involves elleptic eigenvalue problems related to the linearized RDA. This formula was proven by H. Berestycki, F. Hamel and N. Nadirashivili and by H. Weinberger as well in 2002. One important, but subtle, question is whether an incompressible transport term q enhances the spreading or (which means the behavior of c* as a function of the amplitude M of the flow when M is seen as a large parameter). In 2011, A. Zlatoš proved that the speed c* behaves as O(M) when M tends to infinity. In a work of mine, joint with S. Kirsch, we gave the explicit formula for the limit of c*(M)/M as M goes to infinity. In this talk, I will describe this result and will show a sharp criterion on the flow, in 2D, under which we get the optimal speed-up of traveling fronts. I will then show how subtle the question becomes in dimensions 3 or higher. After that, I will discuss a recent result (jointly with S. Kirsch), to appear in Archive for Rational Mechanics and Analysis, about the asymptotics of the KPP minimal speed in 3D flows with ergodic components.

The proof of this result uses PDE tools as well as measure theoretic and functional analytic techniques.

Refreshments will be available at 3:10 p.m.


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