Rate of convergence for periodic homogenization of convex Hamilton–Jacobi equations in one dimension

Abstract

In this paper we study the rate of convergence u ε −→ u as ε −→ 0 + in periodic homogenization of Hamilton-Jacobi equations. Here u ε and u are viscosity solutions to the oscillatory Hamilton-Jacobi equationand its effective equationrespectively. Assuming that the initial data u 0 is bounded and Lipschitz continuous, we provide a simple proof to get the optimal rate O(ε) for a class of Hamiltonians including the classical mechanics one with separable potentialwhere a(·) : R −→ (0, +∞) is continuously differentiable, b(·) : R −→ (−∞, 0] is continuous and 1-periodic.2010 Mathematics Subject Classification. 35B40, 37J50, 49L25 .

Rate of convergence in periodic homogenization for convex Hamilton–Jacobi equations with multiscales

“…The main idea is to break γ 0 into N evenly spaced pieces with respect to time, where N needs to be determined appropriately. For each piece, we approximate its cost by fixing the first argument of L in (12). More precisely, for the kth piece where k = 0, 1, .…”

“…To our best knowledge, the most closely related previous research in this area is [12], where the approach in [8] was extended to attain the optimal rate of O(ϵ) in one dimension with further assumptions on H. In this study, we investigate this problem for dimensions n ⩾ 1 and prove that the convergence rate, in general, is O(t √ ϵ) for t ⩾ √ ϵ.…”

Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with Multiscales

“…To our best knowledge, the most closely related previous research in this area is [11], where the approach in [7] was extended to attain the optimal rate of O(ǫ) in one dimension with further assumptions on H. In this study, we investigate this problem for dimensions n ≥ 1 and prove that the convergence rate, in general, is O( √ ǫ) for t ≥ √ ǫ.…”

On the rate of convergence in homogenization of time-fractional Hamilton-Jacobi equations

“…The argument in [3] can be easily adapted to (1.1) with α = 1 (see [20, Theorem 4.37]). We also refer to [10,18,22] for other development on this subject. In all works [4,10,15,18,21,22], the argument relies on the optimal control formula for u ε , and therefore it is seemingly rather challenging to obtain the optimal rate of convergence of (1.1) for 0 < α < 1, which remains completely open.…”