Uniform exponential contraction for viscous Hamilton–Jacobi equations

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 λ (ν ) characterizing the exponential rate of contraction depends on the viscosity ν . The Markov mechanism provides only a lower bound for λ (ν ) which vanishes in the limit ν→ 0 . At the same time, in the inviscid case ν =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 λ (ν ) which is valid for all ν≥ 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.