论文标题
汉密尔顿的初始价值问题 - 雅各比方程的粘度解决方案，在Pearmons的动力学中发生了归化的汉密尔顿
Viscosity solutions to an initial value problem for a Hamilton--Jacobi equation with a degenerate Hamiltonian occurring in the dynamics of peakons
论文作者
论文摘要
我们考虑了汉密尔顿的初始价值问题 - 雅各比方程，具有二次和堕落的汉密尔顿人。我们的哈密顿量来自Camassa-Holm方程中的$ N $ -Peakon的动力。它是由具有奇异的正式半明确基质的二次形式给出的。这样的问题不属于粘度解决方案的标准理论。同样在我们的情况下，与生存力相关的结果有时用于处理堕落的哈密顿人。我们通过查看相关的最佳控制问题并表明该值函数是粘度解决方案，从而证明了粘度解决方案的全局存在。最复杂的部分是仅在两峰情况下获得的粘度解决方案的连续性。困难的根源是最佳控制问题中解决状态方程的解决方案的非唯一性。我们证明，如果初始条件是Lipschitz连续的，那么粘度解决方案是Lipschitz连续的，并且在短时间内是独一无二的。我们以一个示例结束了论文，显示了粘度解决方案在一维情况下的连续性丧失。
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.
