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

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