We consider an initial value problem for a Hamilton–Jacobi equation with a quadratic and degenerate Hamiltonian. Our Hamiltonian comes from the dynamics of $N$-peakon in the Camassa–Holm equation. It is given by a quadratic form with a singular positive semi-definite matrix. Such a problem does not fall into the standard theory of viscosity solutions. Also viability related results, sometimes used to deal with degenerate Hamiltonians, do not seem applicable in our case. We prove the global existence of a viscosity solution by looking at the associated optimal control problem and showing that the value function is a viscosity solution. The most complicated part is the continuity of a viscosity solution which is obtained in the two-peakon case only. The source of the difficulties is the non-uniqueness of solutions to the state equation in the optimal control problem. We prove that the viscosity solution is Lipschitz continuous and unique on some short time interval if the initial condition is Lipschitz continuous. We end the paper with an example showing the loss of Lipschitz continuity of a viscosity solution in the one-dimensional case.

我们考虑具有二次和简并哈密顿量的哈密顿-雅各比方程的初值问题。我们的哈密顿量来自于$ñ$Camassa–Holm方程中的-peakon。它由具有奇异正半定矩阵的二次形式给出。这样的问题不属于粘度溶液的标准理论。同样，与生存力相关的结果（有时用于处理退化的哈密顿量）似乎不适用于我们的情况。通过查看相关的最优控制问题并证明值函数是粘度解，我们证明了粘度解的全局存在。最复杂的部分是仅在两峰值情况下获得的粘度溶液的连续性。困难的根源在于最优控制问题中状态方程解的非唯一性。我们证明，如果初始条件是Lipschitz连续，则粘度溶液是Lipschitz连续的，并且在较短的时间间隔内是唯一的。我们以一个示例结束本文，该示例显示在一维情况下粘度溶液的Lipschitz连续性损失。