A survey of results on conservation laws with deterministic and random initial data

Abstract

This expository paper examines key results on the dynamics of nonlinear conservation laws with random initial data and applies some theorems to physically important situations. Conservation laws with some nonlinearity, e.g. Burgers' equation, exhibit discontinuous behavior, or shocks, even for smooth initial data. The introduction of randomness in any of several forms into the initial condition renders the analysis extremely complex. Standard methods for tracking a multitude of shock collisions are difficult to implement, suggesting other methods may be needed. We review several perspectives into obtaining the statistics of resulting states and shocks. We present a spectrum of results from a number of works, both deterministic and random. Some of the deep theorems are applied to important discrete examples where the results can be understood in a clearer, more physical context.