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

Shyam Sundar Ghoshal; Parasuram Venkatesh; Emil Wiedemann; Shyam Sundar Ghoshal; Parasuram Venkatesh; Emil Wiedemann

doi:10.3934/nhm.2025046

Networks and Heterogeneous Media

2025, Volume 20, Issue 4: 1061-1086. doi: 10.3934/nhm.2025046

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Research article

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

1.
Centre for Applicable Mathematics, Tata Institute of Fundamental Research, India
2.
Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

Received: 03 June 2025 Revised: 07 August 2025 Accepted: 18 August 2025 Published: 11 October 2025

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.
- scalar conservation law,
- non-local equation,
- shock wave,
- entropy solution,
- convergence to the local model
Citation: Shyam Sundar Ghoshal, Parasuram Venkatesh, Emil Wiedemann. A non-conservative, non-local approximation of the Burgers equation[J]. Networks and Heterogeneous Media, 2025, 20(4): 1061-1086. doi: 10.3934/nhm.2025046

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Abstract

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.

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