On a nonlocal hyperbolic conservation law arising from a gradient constraint problem
Analysis of PDEs
2012-02-07 v1
Abstract
In some models involving nonlinear conservation laws, physical mechanisms exist which prevent the formation of shocks. This gives rise to conservation laws with a constraint on the gradient of the solution. We approach this problem by studying a related conservation law with a spatial nonlocal term. We prove existence, uniqueness and stability of solution of the Cauchy problem for this nonlocal conservation law. In turn, this allows us to provide a notion of solution to the conservation law with a gradient constraint. The proof of existence is based on a time-stepping technique, and an -contraction estimate follows from stability results of Karlsen and Risebro.
Cite
@article{arxiv.1202.1236,
title = {On a nonlocal hyperbolic conservation law arising from a gradient constraint problem},
author = {Paulo Amorim},
journal= {arXiv preprint arXiv:1202.1236},
year = {2012}
}