Mixing in Reaction-Diffusion Systems: Large Phase Offsets

Abstract

We consider Reaction-Diffusion systems on \({\mathbb{R}}\), and prove diffusive mixing of asymptotic states \({u_0(kx - \phi_{\pm}, k)}\), where u0 is a spectrally stable 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 in any topology, 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 is not near the asymptotic profile in any sense. We prove the global existence, decay, and asymptotic self-similarity of the associated wavenumber equation. We develop a functional analytic 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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Acknowledgements

The first author thanks Yan Guo for many stimulating discussions regarding the energy method in Sect. 4. The second author thanks Kevin Zumbrun for discussions during the early stage of this project. Both authors thank the anonymous referees for several useful comments and suggestions. Iyer was supported by the NSF Grants DMS 1611695 and DMS 1209437. Sandstede was partially supported by NSF Grant DMS 1408742.

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Correspondence to Sameer Iyer.

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Iyer, S., Sandstede, B. Mixing in Reaction-Diffusion Systems: Large Phase Offsets. Arch Rational Mech Anal 233, 323–384 (2019). https://doi.org/10.1007/s00205-019-01358-9

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