论文标题
具有两峰汉密尔顿 - 雅各比方程的最佳规律性的粘度解决方案的存在
Existence of viscosity solutions with the optimal regularity of a two-peakon Hamilton--Jacobi equation
论文作者
论文摘要
这项工作致力于对汉密尔顿 - 雅各比方程的研究，其二次和堕落的汉密尔顿学位来自camassa-holm方程中多孔的动力学。它是由具有奇异的正式半明确基质的二次形式给出的。我们增加了上一篇论文中考虑的价值函数的规律性，这是粘度解决方案。我们证明，对于两峰汉密尔顿人来说，这种解决方案实际上是$ 1/2 $-Hölder在太空中连续连续的，而Time-Lipschitz的连续则是连续的。 Time-Lipschitz的规律性在任何维度$ n \ geq 1 $中都得到证明。这种规律性已经在一维的简化中已经知道，而且它是最好的，如我们以前的论文所示。
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 $1/2$-Hölder 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.
