Let H be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton–Jacobi equation \(U_{t}+H(U_x)=0\) and signed Radon measure valued entropy solutions of the conservation law \(u_{t}+[H(u)]_x=0\). After having proved a precise statement of the formal relation \(U_x=u\), we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton–Jacobi equation and signed singular measures in case of the conservation law.

中文翻译：

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Hamilton-Jacobi 方程的不连续解与标量守恒定律的 Radon 测度值解：奇点的消失

令H为有界且 Lipschitz 连续函数。我们考虑 Hamilton-Jacobi 方程\(U_{t}+H(U_x)=0\) 的不连续粘度解和守恒定律\(u_{t}+[H(u)] 的有符号氡测量值熵解_x=0\)。在证明了形式关系\(U_x=u\)的精确陈述之后，我们建立了对解奇点消失的（严格为正的！）时间的估计。这里奇点是哈密顿-雅可比方程中的跳跃不连续点和守恒定律中的带符号奇异测度。