A Non-Conservative, Non-Local Approximation of the Burgers Equation

TL;DR

This paper introduces a novel non-local regularisation for the Burgers equation that provides sharp estimates and insights into the behavior of solutions with discontinuous initial data, advancing understanding of non-local conservation laws.

Contribution

It proposes a new inviscid, non-local regularisation in non-divergence form with sharp a priori estimates and analyzes its limiting behavior for the Burgers equation with discontinuous initial data.

Findings

Sharp a priori estimates on total variation and supremum norm.

Justification of the singular limit for Lipschitz initial data.

Demonstration of non-convergence for initial data with simple discontinuities.

Abstract

The analysis of non-local regularisations of scalar conservation laws is an active research program. Applications of such equations are found in the modelling of physical phenomena such as traffic flow. In this paper, we propose a novel inviscid, non-local regularisation in non-divergence form. The salient feature of our approach is that we can obtain sharp a priori estimates on the total variation and supremum norm, and justify the singular limit for Lipschitz initial data up to the time of catastrophe. For generic conservation laws, this result is sharp, since we can demonstrate non-convergence when the initial data features simple discontinuities. Conservation laws with linear flux derivative, such as the Burgers equation, behave better in the presence of discontinuities. Hence, we devote special attention to the limiting behaviour of non-local solutions with respect to the Burgers equation for a simple class of discontinuous initial data.