Path-dependent convex conservation laws

TL;DR

This paper investigates scalar conservation laws driven by rough paths, demonstrating that solutions are unaffected by replacing the driving path with a piecewise linear approximation, and explores the regularity and properties of these solutions.

Contribution

It establishes the invariance of solutions under piecewise linear approximation of rough paths for convex flux functions in one dimension and analyzes the spatial regularity of solutions.

Findings

Solutions are spatially Lipschitz continuous at certain times.

The solution map's properties depend on the rough path characteristics.

A factorization of the solution operator reveals how the rough path influences solutions.

Abstract

For scalar conservation laws driven by a rough path $z(t)$, in the sense of Lions, Perthame and Souganidis in arXiv:1309.1931, 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, under the assumption of a convex flux function in one spatial dimension. This result is connected to the spatial regularity of solutions. We show that solutions are spatially Lipschitz continuous for a given set of times, depending on the path and the initial data. Fine properties of the map $z \mapsto u(\tau)$, for a fixed time $\tau$, are studied. We provide a detailed description of the properties of the rough path $z(t)$ that influences the solution. This description is extracted by a "factorization" of the solution operator (at time $\tau$). In a companion paper, we make use of the observations herein to construct computationally efficient numerical methods.