In the scalar 1D case, conservation laws and Hamilton–Jacobi equations are deeply related. For both, in the case of a uniformly convex flux/Hamiltonian, we characterize those profiles that can be attained as solutions at a given positive time corresponding to at least one initial datum. Then, for each of the two equations, we precisely identify all those initial data yielding a solution that coincide with a given profile at that positive time. Various topological and geometrical properties of the set of these initial data are then proved.

中文翻译：

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守恒律和汉密尔顿-雅各比方程式中的初始数据识别

在标量一维情况下，守恒律与汉密尔顿-雅各比方程密切相关。对于这两种情况，在均匀凸通量/哈密顿量的情况下，我们表征了可以在给定正时对应于至少一个初始基准的解决方案中获得的轮廓。然后，对于这两个方程式中的每一个，我们都精确地标识出所有初始数据，从而得出与该正时刻给定轮廓一致的解。然后证明了这些初始数据集的各种拓扑和几何特性。