黏性解不存在全局特征
Non-existence of global characteristics for viscosity solutions
摘要：
引入粘性解和变分解两种不同类型的广义解，求解一阶发展Hamilton-Jacobi方程。如果哈密顿量在动量变量中是凸的，则它们重合。在本文中，我们证明了不存在其他类的可积哈密顿共享此属性。为此，我们建立任何非凸非凹可积哈密顿一个光滑的初始条件，使图的黏性解不包含在波前与柯西问题。该结构是基于一个新的例子鞍哈密顿和精确的分析一维的情况下，再加上减少和近似参数。
英文摘要：
Two different types of generalized solutions, namely viscosity and variational solutions, were introduced to solve the first-order evolutionary Hamilton--Jacobi equation. They coincide if the Hamiltonian is convex in the momentum variable. In this paper we prove that there exists no other class of integrable Hamiltonians sharing this property. To do so, we build for any non-convex non-concave integrable Hamiltonian a smooth initial condition such that the graph of the viscosity solution is not contained in the wavefront associated with the Cauchy problem. The construction is based on a new example for a saddle Hamiltonian and a precise analysis of the one-dimensional case, coupled with reduction and approximation arguments.
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