Contractive Metrics for Nonsmooth Evolutions

Given an evolution equation, a standard way to prove the well posedness of the Cauchy problem is to establish a Gronwall type estimate, bounding the distance between any two trajectories. There are important cases, however, where such estimates cannot hold, in the usual distance determined by the Euclidean norm or by a Banach space norm. In alternative, one can construct different distance functions, related to a Riemannian structure or to an optimal transportation problem. This paper reviews various cases where this approach can be implemented, in connection with discontinuous ODEs on IRn, nonlinear wave equations, and systems of conservation laws. For all the evolution equations considered here, a metric can be constructed such that the distance between any two solutions satisfies a Gronwall type estimate. This yields the uniqueness of solutions, and estimates on their continuous dependence on the initial data.