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

**Authors:** Shyam Sundar Ghoshal, Parasuram Venkatesh, Emil Wiedemann  
**arXiv:** [2410.16743](https://arxiv.org/abs/2410.16743)  
**Date:** 2024-10-22  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2410.16743v1/

## Key points

- Authors: Shyam Sundar Ghoshal, Parasuram Venkatesh, and Emil Wiedemann
- Study focus: Non-local regularisations of scalar conservation laws
- Novel approach: Proposing inviscid, non-local regularisation in non-divergence form
- Derivation of precise a priori estimates on total variation and supremum norm
- Justification of singular limit for Lipschitz initial data up to point of catastrophe
- Showcase instances of non-convergence for generic conservation laws with simple discontinuities in initial data
- Conservation laws with linear flux derivatives exhibit more favorable behavior with discontinuities
- Exploration: Limiting behavior of non-local solutions for Burgers equation with specific class of discontinuous initial data
- Contribution: Advancing understanding in non-local regularisations field and insights on conservation laws' response to discontinuities

## Summary

In their paper titled "A Non-Conservative, Non-Local Approximation of the Burgers Equation," authors Shyam Sundar Ghoshal, Parasuram Venkatesh, and Emil Wiedemann delve into the analysis of non-local regularisations of scalar conservation laws. This is a vibrant area of research with applications in modeling physical phenomena like traffic flow. The focus of their study is on proposing a novel inviscid, non-local regularisation in non-divergence form. One key aspect of their approach is the ability to derive precise a priori estimates on the total variation and supremum norm while also justifying the singular limit for Lipschitz initial data up to the point of catastrophe. The authors demonstrate that for generic conservation laws, their results are particularly significant as they can showcase instances of non-convergence when dealing with initial data containing simple discontinuities. They also highlight that conservation laws featuring linear flux derivatives exhibit more favorable behavior in the presence of discontinuities. As a result, special attention is dedicated to exploring the limiting behavior of non-local solutions concerning the Burgers equation for a specific class of discontinuous initial data. This comprehensive study not only contributes to advancing understanding in the field of non-local regularisations but also sheds light on how different types of conservation laws respond to discontinuities in their initial data. By providing rigorous analysis and insightful observations, Ghoshal, Venkatesh, and Wiedemann offer valuable insights that can potentially inform further research and development in this area.