We consider the scalar conservation law in one space dimension with a genuinely nonlinear flux. We assume that an appropriate velocity function depending on the entropy solution of the conservation law is given for the comprising particles, and study their corresponding trajectories under the flow. The differential equation that each of these trajectories satisfies depends on the entropy solution of the conservation law which is typically discontinuous in both time and space variables. The existence and uniqueness of these trajectories are guaranteed by the Filippov theory of differential equations. We show that such a Filippov solution is compatible with the front tracking and vanishing viscosity approximations in the sense that the approximate trajectories given by either of these methods converge uniformly to the trajectories corresponding to the entropy solution of the scalar conservation law. For certain classes of flux functions, illustrated by traffic flow, we prove the H\"older continuity of the particle trajectories with respect to the initial field or the flux function. We then consider the inverse problem of recovering the initial field or the flux function of the scalar conservation law from discrete pointwise measurements of the particle trajectories. We show that the above continuity properties translate to the stability of the Bayesian regularised solutions of these inverse problems with respect to appropriate approximations of the forward map. We also discuss the limitations of the situation where the same inverse problems are considered with pointwise observations made from the entropy solution itself.

翻译：我们考虑一维空间中具有严格非线性通量的标量守恒律。假设给定一个依赖于守恒律熵解的合适速度函数用于描述组成粒子，我们研究这些粒子在流作用下的相应轨迹。每个轨迹满足的微分方程依赖于守恒律的熵解，而该熵解在时间和空间变量上通常是不连续的。这些轨迹的存在性和唯一性由菲利波夫微分方程理论保证。我们证明此类菲利波夫解与前向追踪和粘性消失近似方法兼容，即这两种方法给出的近似轨迹一致收敛于标量守恒律熵解对应的轨迹。对于通量函数的特定类别（以交通流为例），我们证明粒子轨迹关于初始场或通量函数的赫尔德连续性。随后考虑从粒子轨迹的离散点观测中恢复标量守恒律初始场或通量函数的反问题。我们证明上述连续性性质可转化为这些反问题的贝叶斯正则化解关于前向映射适当近似的稳定性。同时讨论在直接对熵解本身进行点观测时，相同反问题所面临的情形限制。