Nonlinear scalar conservation laws are traditionally viewed as transport equations. We take instead the viewpoint of these PDEs as continuity equations with an implicitly defined velocity field. We show that a weak solution is the entropy solution if and only if the ODE corresponding to its velocity field is well-posed. We also show that the flow of the ODE is 1/2-Hölder regular. Finally, we give several examples showing that our results are sharp, and we provide explicit computations in the case of a Riemann problem.

中文翻译：

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双曲守恒定律的粒子路径

非线性标量守恒定律传统上被视为输运方程。相反，我们将这些偏微分方程视为具有隐式定义的速度场的连续性方程。我们证明，当且仅当与其速度场对应的 ODE 适定时，弱解才是熵解。我们还表明 ODE 的流程为$1/2$-Hölder 常规。最后，我们给出了几个例子，表明我们的结果很清晰，并且我们在黎曼问题的情况下提供了显式计算。