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

TL;DR

This paper proves that viscosity solutions to a two-peakon Hamilton--Jacobi equation exhibit optimal regularity, specifically 1/2-Hölder continuity in space and Lipschitz continuity in time, extending known results to higher dimensions.

Contribution

It establishes the optimal regularity of viscosity solutions for a multipeakon Hamilton--Jacobi equation with a degenerate quadratic Hamiltonian in any dimension.

Findings

Viscosity solutions are 1/2-Hölder continuous in space.

Solutions are Lipschitz continuous in time for all dimensions.

Optimal regularity results extend previous one-dimensional findings.

Abstract

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.