Nonlinear Stability through Algebraically Decaying Point Spectrum: Applications to Nonlocal Interaction Equations

Abstract

Nonlocal interaction equations, such as aggregation equations and a number of related models for biological swarming, can exhibit compact, co-dimension one equilibrium solutions. Nonlinear stability of these solutions crucially depends on understanding the properties of operators whose linearizations have point spectrum that accumulates on the imaginary axis. In this paper, we establish criteria upon the linear operator, the nonlinearity, and the admissible perturbations under which linear stability is sufficient to guarantee nonlinear stability in the presence of such algebraically decaying point spectrum. We also provide temporal rates at which admissible perturbations decay to zero. We then apply the theory to some simplified nonlocal equations. tions [10, 11, 16, 17, 32, 41, 43] and related models for biological swarming [18,31,44,46,53,54], support stationary solutions where the entire particle density concentrates on a co-dimension one manifold [2,3,4,9,35,36,42,52,55,56,57]. The nonlocal aggregation equation in R d , Introduction. Many nonlocal interaction equations, such as aggregation equa

Anisotropic assembly and pattern formation

“…Our analytical approach requires the formulation of gradient-flow dynamics for the particlebased energy (1) as well as a subsequent continuum approximation of these dynamics. This parallels the approach developed in [30,51] for the isotropic case, where a Birkhoff-Rott type reformulation [48] of the well-studied aggregation equation [1,2,5,6,8,11,31,49] provides the requisite continuum description. As we shall pursue a similar strategy, we therefore derive the analogous set of equations in the anisotropic case.…”

“…Thus ( ) σ A always has at least one eigenvalue decaying to zero in the high-frequency limit. High-frequency stability (and hence stability) therefore necessitates algebraically decaying point spectrum in the sense of [49], in that the corresponding linear operator L g,iso has point spectrum that accumulates on the imaginary axis from below.…”