Open Access
March/April 2013 Quasi-gradient systems, modulational dichotomies, and stability of spatially periodic patterns
Alin Pogan, Arnd Scheel, Kevin Zumbrun
Differential Integral Equations 26(3/4): 389-438 (March/April 2013). DOI: 10.57262/die/1360092829


Extending the approach of Grillakis, Shatah, and Strauss, Bronski, Johnson, and Kapitula, and others for Hamiltonian systems, we explore relations between the constrained variational problem $$ \min_{X:C(X)=c_0} \mathcal{E}(X) , \ \ \ c_0\in \mathbb R^r , $$ and stability of solutions of a class of degenerate ``quasi-gradient'' systems $dX/dt=-M(X)\nabla \mathcal{E}(X)$ admitting constraints, including Cahn--Hilliard equations, one- and multi-dimensional viscoelasticity, and coupled conservation-law--reaction-diffusion systems arising in chemotaxis and related settings. Using the relation between variational stability and the signature of $\partial c/\partial \omega \in \mathbb R^{r\times r}$, where $c(\omega)=C(X^*_\omega)\in \mathbb R ^r$ denote the values of the imposed constraints and $\omega\in \mathbb R^r$ the associated Lagrange multipliers at a critical point $X^*_\omega$, we obtain as in the Hamiltonian case a general criterion for co-periodic stability of periodic waves, illuminating and extending a number of previous results obtained by direct Evans-function techniques. More interestingly, comparing the form of the Jacobian arising in the co-periodic theory to Jacobians arising in the formal Whitham equations associated with modulation, we recover and substantially generalize a previously mysterious ``modulational dichotomy'' observed in special cases by Oh--Zumbrun and Howard, showing that co-periodic and sideband stability are incompatible. In particular, we both illuminate and extend to general viscosity/strain-gradient effects and multidimensional deformations the result of Oh-Zumbrun of universal modulational instability of periodic solutions of the equations of viscoelasticity with strain-gradient effects, considered as functions on the whole line. Likewise, we generalize to multi-dimensions corresponding results of Howard on periodic solutions of Cahn--Hilliard equations.


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Alin Pogan. Arnd Scheel. Kevin Zumbrun. "Quasi-gradient systems, modulational dichotomies, and stability of spatially periodic patterns." Differential Integral Equations 26 (3/4) 389 - 438, March/April 2013.


Published: March/April 2013
First available in Project Euclid: 5 February 2013

zbMATH: 1289.35030
MathSciNet: MR3059169
Digital Object Identifier: 10.57262/die/1360092829

Primary: 35B10 , 35B35

Rights: Copyright © 2013 Khayyam Publishing, Inc.

Vol.26 • No. 3/4 • March/April 2013
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