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

This work is devoted to the studies of 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 our previous paper, which is known to be the viscosity solution. We prove that for a two-peakon Hamiltonian such solutions are actually $1/2$-H\"{o}lder continuous in space and time-Lipschitz continuous. The time-Lipschitz regularity is proven in any dimension $N\geq 1$. Such a regularity is already known in the one-dimensional simplifications, moreover it is the best possible, as was shown in our previous papers.