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

Abstract

Abstract We investigate the asymptotic behavior of solutions of Hamilton–Jacobi equations with large Hamiltonian drift terms in an open subset of the two-dimensional Euclidean space. The drift is given by ε - 1 ⁢ ( H x 2 , - H x 1 ) {{\varepsilon}^{-1}(H_{x_{2}},-H_{x_{1}})} of a Hamiltonian H, with ε > 0 {{\varepsilon}>0} . We establish the convergence, as ε → 0 + {{\varepsilon}\to 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 more than four segments at a node.