A non-conservative, non-local approximation of the Burgers equation

The analysis of non-local regularisations of scalar conservation laws is an active research program. Applications of such equations are found in modelling physical phenomena, such as traffic flow. In this paper, we propose an inviscid, non-local regularisation in a non-divergence form. The salient feature of our approach is that we can obtain sharp a priori estimates on the total variation $(TV)$ and supremum norm and justify the singular limit for Lipschitz initial data up to the time of catastrophe. For generic conservation laws, this result was sharp, since we could demonstrate non-convergence when the initial data featured simple discontinuities. However, when the flux derivative was linear, such as for the Burgers equation, we obtained stronger limits on the singular limit. Therefore, we devoted special attention to regularisations of the Burgers equation, specifically the limiting behaviour of solutions to the Cauchy problems with fixed initial data.