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

Abstract

This work, which continues the research begun in [6], [7], 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 [6], which is known to be the viscosity solution. We prove that for a two-peakon Hamiltonian such solutions are actually 1/2-Hölder continuous in space and time-Lipschitz continuous. The time-Lipschitz regularity is proven in any dimension N ≥ 1. Such a regularity is already known in the one-dimensional simplifications (see [7]), moreover it is the best possible, as shown in [6,7].