Mixing in Reaction-Diffusion Systems: Large Phase Offsets

TL;DR

This paper proves diffusive mixing of asymptotic states in reaction-diffusion systems with large phase offsets, establishing global stability, existence, decay, and self-similarity results using a novel functional framework and coercivity estimates.

Contribution

It introduces a new analysis for reaction-diffusion systems with large phase offsets, handling arbitrarily large data and proving global stability and decay.

Findings

Global existence and decay of solutions

Asymptotic self-similarity established

New coercivity estimate for linearized operator

Abstract

We consider Reaction-Diffusion systems on $\mathbb{R}$, and prove diffusive mixing of asymptotic states $u_0(kx - \phi_{\pm}, k)$, where $u_0$ is a periodic wave. Our analysis is the first to treat arbitrarily large phase-offsets $\phi_d = \phi_{+}- \phi_{-}$, so long as this offset proceeds in a sufficiently regular manner. The offset $\phi_d$ completely determines the size of the asymptotic profiles, placing our analysis in the large data setting. In addition, the present result is a global stability result, in the sense that the class of initial data considered are not near the asymptotic profile in any sense. We prove global existence, decay, and asymptotic self-similarity of the associated wavenumber equation. We develop a functional framework to handle the linearized operator around large Burgers profiles via the exact integrability of the underlying Burgers flow. This framework enables us to prove a crucial, new mean-zero coercivity estimate, which we then combine with a nonlinear energy method.