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We consider an initial value problem for a Hamilton--Jacobi equation with aquadratic and degenerate Hamiltonian. Our Hamiltonian comes from the dynamicsof $N$-peakon in the Camassa--Holm equation. It is given by a quadratic formwith a singular positive semi-definite matrix. Such a problem does not fallinto the standard theory of viscosity solutions. Also viability relatedresults, sometimes used to deal with degenerate Hamiltonians, do not seemapplicable in our case. We prove the global existence of a viscosity solutionby looking at the associated optimal control problem and showing that the valuefunction is a viscosity solution. The most complicated part is the continuityof a viscosity solution which is obtained in the two-peakon case only. Thesource of the difficulties is the non-uniqueness of solutions to the stateequation in the optimal control problem. We prove that the viscosity solutionis Lipschitz continuous and unique on some short time interval if the initialcondition is Lipschitz continuous. We end the paper with an example showing theloss of Lipschitz continuity of a viscosity solution in the one-dimensionalcase.

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