Existence of solutions to a one-dimensional Hamilton–Jacobi equation with a degenerate Hamiltonian

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

“…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].…”

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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.

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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

“…However, it is not expected that viscosity solutions to (1.1) admit such a high regularity. Indeed, in [6] explicit formulas for viscosity solutions to the one-dimensional simplification of (1.1) are found. One computes that they are exactly 1/2-Hölder continuous in space variable.…”

“…The two-dimensional case, being the subject of the present paper, is a first step to be made. Let us also mention the one-dimensional simplifications of (1.1) which are successfully studied in [6]. However, as noticed in [6], the methods used there are peculiar to the one-dimensional setting.…”

“…Let us also mention the one-dimensional simplifications of (1.1) which are successfully studied in [6]. However, as noticed in [6], the methods used there are peculiar to the one-dimensional setting. The higher-dimensional case requires more advanced approach.…”