Uniform exponential contraction for viscous Hamilton-Jacobi equations

TL;DR

This paper proves a uniform exponential contraction rate for viscous Hamilton-Jacobi equations with kicked forcing, unifying the dynamical and Markov mechanisms across all viscosities including the inviscid limit.

Contribution

It establishes a uniform lower bound for the Lyapunov exponent in the viscous Hamilton-Jacobi equation with kicked forcing, combining PDE and Weak KAM theory methods.

Findings

Uniform lower bound for Lyapunov exponent for all viscosities

Unified dynamical and Markov mechanisms for exponential contraction

Applicable to discrete time viscous Hamilton-Jacobi equations

Abstract

The well known phenomenon of exponential contraction for solutions to the viscous Hamilton-Jacobi equation in the space-periodic setting is based on the Markov mechanism. However, the corresponding Lyapunov exponent $\lambda(\nu)$ characterizing the exponential rate of contraction depends on the viscosity $\nu$. The Markov mechanism provides only a lower bound for $\lambda(\nu)$ which vanishes in the limit $\nu \to 0$. At the same time, in the inviscid case $\nu=0$ one also has exponential contraction based on a completely different dynamical mechanism. This mechanism is based on hyperbolicity of action-minimizing orbits for the related Lagrangian variational problem. In this paper we consider the discrete time case (kicked forcing), and establish a uniform lower bound for $\lambda(\nu)$ which is valid for all $\nu\geq 0$. The proof is based on a nontrivial interplay between the dynamical and Markov mechanisms for exponential contraction. We combine PDE methods with the ideas from the Weak KAM theory.