Path-dependent convex conservation laws

For convex scalar conservation laws in 1-d, driven by a rough path z(t), in the sense of Lions, Perthame and Souganidis in [32], we show that it is possible to replace z(t) by a piecewise linear path, and still obtain the same solution at a given time. This result is connected to the spatial regularity of solutions. We show that solutions are spatially Lipschitz continuous for an a priori set of times, depending on the path and the initial data. Fine properties of the map z↦u(τ), for a prescribed time τ, are studied. We provide a detailed description of the properties of the rough path z(t) that influence the solution. This description is extracted by a “factorization” of the solution operator (at time τ). In a companion paper [24], we make use of the observations herein to construct computationally efficient numerical methods.