Abstract:
I will discuss two recent papers [1, 2] on the reaction-diffusion system with nonmonotone reaction function F and one non-diffusing component. As speed of reaction tends to infinity, the concentration of the non-diffusing component exhibits fast oscillations. We identify precisely its Young measure which, as a by-product, proves strong convergence of the diffusing component, a result that is not obvious at all from a priori estimates. Our work is based on analysis of regularization for forward-backward parabolic equations by Plotnikov [3]. 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. Finally, in [2] we upgrade the method of Plotnikov by application of classical Radon-Nikodym theorem.

This is a joint work with Benoît Perthame (Sorbonne University, Paris).

References:
[1] B. Perthame, J. Skrzeczkowski. Fast reaction limit with nonmonotone reaction function. To appear in Communications on Pure and Applied Mathematics, arXiv: 2008.11086.
[2] J. Skrzeczkowski, Fast reaction limit and forward-backward diffusion: a Radon-Nikodym approach. arXiv preprint: 2105.11218.
[3] P. I. Plotnikov. Passage to the limit with respect to viscosity in an equation with a variable direction of parabolicity. Differ. Uravn., 30:4 (1994), 665674; Differ. Equ., 30:4 (1994), 614622.