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We consider the scalar conservation law in one space dimension with agenuinely nonlinear flux. We assume that an appropriate velocity functiondepending on the entropy solution of the conservation law is given for thecomprising particles, and study their corresponding trajectories under theflow. The differential equation that each of these trajectories satisfiesdepends on the entropy solution of the conservation law which is typicallydiscontinuous in both time and space variables. The existence and uniqueness ofthese trajectories are guaranteed by the Filippov theory of differentialequations. We show that such a Filippov solution is compatible with the fronttracking and vanishing viscosity approximations in the sense that theapproximate trajectories given by either of these methods converge uniformly tothe trajectories corresponding to the entropy solution of the scalarconservation law. For certain classes of flux functions, illustrated by trafficflow, we prove the H\"older continuity of the particle trajectories withrespect to the initial field or the flux function. We then consider the inverseproblem of recovering the initial field or the flux function of the scalarconservation law from discrete pointwise measurements of the particletrajectories. We show that the above continuity properties translate to thestability of the Bayesian regularised solutions of these inverse problems withrespect to appropriate approximations of the forward map. We also discuss thelimitations of the situation where the same inverse problems are consideredwith pointwise observations made from the entropy solution itself.

Stability of particle trajectories of scalar conservation laws and applications in Bayesian inverse problems