Work

Existence and higher regularity of statistically steady states for the stochastic Coleman-Gurtin equation

Year: 2025

Type: article

Abstract: We study a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. We show that for a broad class of non-linear potentials, the system always admits invariant probability measures. However, the presence of memory effects precludes access to compactness in a typical fashion. In this paper, this obstacle is overcome by introducing functional spaces adapted to the memory kernels, thereby allowing one to recover compactness. Under the assumption of sufficiently smooth noise, it is then shown that the statistically stationary states possess higher-order regularity properties dictated by the structure of the nonlinearity. This is established through a control argument that asymptotically transfers regularity onto the solution by exploiting the underlying Lyapunov structure of the system in a novel way.

Source: Evolution equations and control theory

DOI: 10.3934/eect.2025038

Authors: Nathan E. Glatt-Holtz, Vincent R. Martinez, Hung D. Nguyen

Institutions: Indiana University Bloomington, Indiana University, Hunter College, University of Tennessee at Knoxville

Language: English

Topic: Stability and Controllability of Differential Equations

Subfield: Control and Systems Engineering

Field: Engineering

Domain: Physical Sciences

Open Access status: diamond

Funders: National Science Foundation, City University of New York

Awards: Stochastic Methods in Fluid Mechanics: Ergodic Properties, Statistical Sampling, and Uncertainty Quantification, LEAPS-MPS: Dynamical Parameter Estimation for Hydrodynamic Equations, Analysis of Nonlinear Stochastic Partial Differential Equations with Applications in Turbulence Theory and Climate Modeling, Collaborative Research: Mathematical Analysis of the Effects of Rotation, Stratification, and Dissipation in Incompressible Fluid Flows, Collaborative Research: Bayesian Inversion Approaches to Partial Differential Equations: Theory, Algorithm Development, and Applications