Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity

Abstract

We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be reformulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the sublinear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.

Random homogenization of coercive Hamilton–Jacobi equations in 1d

“…If H(p, x, ω) is convex with respect to p ∈ R d , stochastic homogenization was proved independently by Souganidis [9] and by Rezakhanlou-Tarver [7]. This result was extended to tdependent case by Schwab [8] when the Hamiltonian has super-linear growth in p and by Jing-Souganidis-Tran [6] for Hamiltonians with the form a(x, t, ω)|p|. For those quasi-convex Hamiltonians, Siconolfi and Davini [5] established the random homogenization in 1d, and the general dimensional case was proved by Amstrong-Souganidis [2].…”

Feeble Fish in Time‐Dependent Waters and Homogenization of the G‐equation

“…If V t is periodic in x and random, statistically stationary, and ergodic with respect to t, then the homogenization limit can be proven by an argument given in [9].…”

Stochastic homogenization of viscous superquadratic Hamilton–Jacobi equations in dynamic random environment

“…For firstorder equations with superlinear Hamiltonians, homogenization results were proved by Schwab [26]. Recently, the authors [17] established homogenization for linearly growing Hamiltonians that are periodic in space and stationary ergodic in time.…”