Viscosity solutions to an initial value problem for a Hamilton–Jacobi equation with a degenerate Hamiltonian occurring in the dynamics of peakons

Abstract We consider an initial value problem for a Hamilton–Jacobi equation with quadratic and degenerate Hamiltonian. Our Hamiltonian comes from the dynamics of N -peakon in Camassa–Holm equation. It is given by form singular positive semi-definite matrix. Such does not fall into standard theory viscosity solutions. Also viability related results, sometimes used to deal Hamiltonians, do seem applicable our case. prove global existence solution looking at associated optimal control showing that function solution. The most complicated part continuity which obtained two-peakon case only. source difficulties non-uniqueness solutions state problem. Lipschitz continuous unique on some short time interval if condition continuous. end paper example loss one-dimensional