Quasi-gradient systems, modulational dichotomies, and stability of spatially periodic patterns
Quasi-gradient systems, modulational dichotomies, and stability of spatially periodic patterns
复制标题
准梯度系统、调制二分法和空间周期模式的稳定性
DOI：
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发表时间：
2012
影响因子：
1.4
通讯作者：
K. Zumbrun
中科院分区：
文献类型：
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作者：
A. Pogan;A. Scheel;K. Zumbrun
扩展了 Grillakis-Shatah-Strauss、Bronski-Johnson-Kapitula 等人针对哈密顿系统的方法，我们探讨了约束变分问题 $min_{X:C(X)=c_0} mathcal{E}(X)$、$c_0in RM^r$ 与一类退化“准梯度”系统 $dX/dt=-M(X) 解的稳定性之间的关系
abla mathcal{E}(X)$ 接受约束，包括 Cahn-Hilliard 方程、一维和多维粘弹性以及趋化性和相关设置中出现的耦合守恒定律-反应扩散系统。利用变分稳定性与 R^{r imes r}$ 中 $partial c/partial omega 的签名之间的关系，其中 RM^r$ 中的 $c(omega)=C(X^*_omega)in RM^r$ 表示施加的约束值，$omegain RM^r$ 表示临界点 $X^*_omega$ 处相关的拉格朗日乘子，我们在哈密顿情况下获得了周期波同周期稳定性的一般准则，阐明并扩展了许多先前的准则通过直接埃文斯函数技术获得的结果。更有趣的是，将同周期理论中出现的雅可比行列式的形式与与调制相关的正式惠特姆方程中出现的雅可比行列式进行比较，我们恢复并基本上概括了Oh-Zumbrun和Howard在特殊情况下观察到的先前神秘的“调制二分法”，表明同周期和边带稳定性是不相容的。特别是，我们阐明并扩展到一般粘度/应变梯度效应和多维变形，具有应变梯度效应的粘弹性方程的周期解的普遍调制不稳定性的 Oh-Zumbrun 结果，被视为整条线上的函数。同样，我们将 Howard 对 Cahn-Hilliard 方程周期解的相应结果推广到多维。
Extending the approach of Grillakis-Shatah-Strauss, Bronski-Johnson-Kapitula, and others for Hamiltonian systems, we explore relations between the constrained variational problem $min_{X:C(X)=c_0} mathcal{E}(X)$, $c_0in RM^r$, and stability of solutions of a class of degenerate "quasi-gradient" systems $dX/dt=-M(X)
abla mathcal{E}(X)$ 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 $partial c/partial omega in R^{r imes r}$, where $c(omega)=C(X^*_omega)in RM^r$ denote the values of the imposed constraints and $omegain RM^r$ 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.