Mixing in Reaction-Diffusion Systems: Large Phase Offsets

Sameer Iyer; Bjorn Sandstede

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arXiv · 1610.06527

Mixing in Reaction-Diffusion Systems: Large Phase Offsets

Sameer Iyer·Bjorn Sandstede

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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.

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Sameer Iyer, Bjorn Sandstede. 2016-10-20. Mixing in Reaction-Diffusion Systems: Large Phase Offsets. https://arxiv.org/abs/1610.06527

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