Differential and Integral Equations

Quasi-gradient systems, modulational dichotomies, and stability of spatially periodic patterns

Alin Pogan, Arnd Scheel, and Kevin Zumbrun

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Abstract

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.

Article information

Source
Differential Integral Equations, Volume 26, Number 3/4 (2013), 389-438.

Dates
First available in Project Euclid: 5 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1360092829

Mathematical Reviews number (MathSciNet)
MR3059169

Zentralblatt MATH identifier
1289.35030

Subjects
Primary: 35B35: Stability 35B10: Periodic solutions

Citation

Pogan, Alin; Scheel, Arnd; Zumbrun, Kevin. Quasi-gradient systems, modulational dichotomies, and stability of spatially periodic patterns. Differential Integral Equations 26 (2013), no. 3/4, 389--438. https://projecteuclid.org/euclid.die/1360092829


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