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Archive for Rational Mechanics and Analysis

, Volume 233, Issue 1, pp 323–384 | Cite as

Mixing in Reaction-Diffusion Systems: Large Phase Offsets

  • Sameer IyerEmail author
  • Björn Sandstede
Article
  • 86 Downloads

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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Notes

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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Conflict of interest

The authors declare that they have no conflicts of interest.

References

  1. 1.
    Bricmont J., Kupiainen A., Lin G.: Renormalization Group and Asymptotics of Solutions of Nonlinear Parabolic Equations. Commun. Pure. Appl. Math. 47, 893–922 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Doelman, A., Sandstede, B., Scheel, A., Schneider, G.: The dynamics of modulated wave trains, Mem. Am. Math. Soc. 199(934) 2009Google Scholar
  3. 3.
    Folland, G.: Real Analysis. Wiley. Second EditionGoogle Scholar
  4. 4.
    Goodman J., Szpessy A., Zumbrun K.: A remark on the stability of viscous shock waves. SIAM J. Math. Anal. 6(25, 1463–1467), 6 25, 1463–1467 (1994)MathSciNetGoogle Scholar
  5. 5.
    Johnson M., Noble P., Rodrigues L.M., Zumbrun K.: Nonlocalized modulation of periodic reaction diffusion waves: nonlinear stability. Arch. Rational Mech. Anal. 207, 693–715 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Johnson M., Noble P., Rodrigues L.M., Zumbrun K.: Nonlocalized modulation of periodic reaction diffusion waves: the Whitham equation. Arch. Rational Mech. Anal. 207, 669–692 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Johnson M., Zumbrun K.: Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations. Ann. Inst. H. Poincare Anal. Non Lineaire 28(4), 499–527 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Miller J., Bernoff A.: Rates of convergence to self-similar solutions of Burgers’ equation. Stud. Appl. Math. 111, 29–40 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sandstede B., Scheel A., Schneider G., Uecker H.: Diffusive mixing of periodic wave trains in Reaction-Diffusion systems. J. Differ. Equs. 252, 3541–3574 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

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