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Fast reaction limit with nonmonotone reaction function

Banff International Research Station for Mathematical Innovation and Discovery

We analyse fast reaction limit in the reaction-diffusion system \begin{align*} \partial_t u^{\varepsilon} &= \frac{v^{\varepsilon} - F(u^{\varepsilon})}{\varepsilon}, \\ \partial_t v^{\varepsilon} &= \Delta v^{\varepsilon} + \frac{F(u^{\varepsilon}) - v^{\varepsilon}}{\varepsilon}, \end{align*} with nonmonotone reaction function $F$. As speed of reaction tends to infinity, the concentration of non-diffusing component $u^{\varepsilon}$ exhibits fast oscillations. We identify precisely its Young measure which, as a by-product, proves strong convergence of the diffusing component $v^{\varepsilon}$, a result that is not obvious from a priori estimates. Our work is based on analysis of regularization for forward-backward parabolic equations by Plotnikov [2]. We rewrite his ideas in terms of kinetic functions which clarifies the method, brings new insights, relaxes assumptions on model functions and provides a weak formulation for the evolution of the Young measure. </p>

This is a joint work with Beno\^\i t Perthame (Paris) [1] </p>

[1] B. Perthame, J. Skrzeczkowski. <i>Fast reaction limit with nonmonotone reaction function</i>. arXiv: 2008.11086, submitted.
[2] P. I. Plotnikov. <i>Passage to the limit with respect to viscosity in an equation with a variable direction of parabolicity.</i> Differ. Uravn., 30:4 (1994), 665--674; Differ. Equ., 30:4 (1994), 614--622. </p>

Mathematics; Partial differential equations; Fluid mechanics; Fluid dynamics

video/mp4

Author affiliation: University of Warsaw

2021-05-27

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/78466

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/