Mixing in Reaction-Diffusion Systems: Large Phase Offsets

2016/10/20 by Iyer, Sameer, Sandstede, Bjorn
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1610.06527

Abstract

We consider Reaction-Diffusion systems on ℝ, and prove diffusive mixing of asymptotic states u₀(kx - ϕ_±, k), where u₀ is a periodic wave. Our analysis is the first to treat arbitrarily large phase-offsets ϕ_d = ϕ₊- ϕ_-, so long as this offset proceeds in a sufficiently regular manner. The offset ϕ_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.

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