On the long-time behavior of a perturbed conservative system with degeneracy

TL;DR

This paper analyzes the long-term behavior of a degenerate conservative system under dissipation and stochastic perturbations, revealing how the system evolves as perturbations diminish, with implications for turbulence-related PDEs.

Contribution

It characterizes the asymptotic behavior of a degenerate conservative system with stochastic perturbations, linking degeneracy to turbulence-like features.

Findings

Long-time limit characterized as perturbation tends to zero

Degeneracy leads to features similar to turbulence in PDEs

Provides insights into the stability of perturbed conservative systems

Abstract

We consider in this work a model conservative system subject to dissipation and Gaussian-type stochastic perturbations. The original conservative system possesses a continuous set of steady states, and is thus degenerate. We characterize the long-time limit of our model system as the perturbation parameter tends to zero. The degeneracy in our model system carries features found in some partial differential equations related, for example, to turbulence problems.