An asymptotic analysis for Hamilton–Jacobi equations with large Hamiltonian drift terms

Abstract

We investigate the asymptotic behavior of solutions of Hamilton-Jacobi equations with large drift term in an open subset of two-dimensional Euclidean space. When the drift is given by ε −1 (H x 2 , −H x 1 ) of a Hamiltonian H, with ε > 0, we establish the convergence, as ε → 0+, of solutions of the Hamilton-Jacobi equations and identify the limit of the solutions as the solution of systems of ordinary differential equations on a graph. This result generalizes the previous one obtained by the author to the case where the Hamiltonian H admits a degenerate critical point and, as a consequence, the graph may have segments more than four at a node.

Averaging of Hamilton-Jacobi equations along divergence-free vector fields

“…The geometry of H is stated as follows (see also [14]). The set D 0 = {x ∈ R 2 | H(x) > 0} is open and connected, and the open set {x ∈ R 2 | H(x) < 0} has exactly N − 1 connected components D i , with i ∈ I 1 , such that z i ∈ D i (see Figure 1).…”

“…A crucial difference of this work from [13,14] is that G is not anymore convex so that the results cover the differential games processes. Another critical point here is that we treat the Dirichlet boundary condition in the viscosity sense, which makes the statement of our results transparent.…”

“…There are two difficulties to be dealt with here beyond those in [13,14]. One is that the optimal control interpretation is not available anymore of the problem, and the second is how to deal with the boundary layer and to determine the effective boundary data.…”