Existence of viscosity solutions with the optimal regularity of a two-peakon Hamilton–Jacobi equation

We study here a Hamilton–Jacobi equation with a quadratic and degenerate Hamiltonian, which comes from the dynamics of a multipeakon in the Camassa–Holm equation. It is given by a quadratic form with a singular positive semi-definite matrix. We increase the regularity of the value function considered in earlier works, which is known to be the viscosity solution. We prove that for a two-peakon Hamiltonian such solutions are actually [Formula: see text]-Hölder continuous in space and time-Lipschitz continuous. The time-Lipschitz regularity is proven in any dimension [Formula: see text]. Such a regularity is already known in the one-dimensional case and, moreover it is the best possible, as shown earlier.