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Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws

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Abstract

This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws. The basic idea is that the “meaningful objects” are the fluxes, evaluated across domain boundaries over time intervals. The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting. It implies that a weak solution indeed satisfies the balance law. In fact, it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary. It should be emphasized that the weak solutions considered here need not be entropy solutions. Furthermore, the assumption imposed on the flux f(u) is quite minimal—just that it is locally bounded.

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Acknowledgements

The first author thanks the Institute of Applied Physics and Computational Mathematics, Beijing, for the hospitality and support. The second author is supported by the NSFC (Nos. 11771054, 12072042,91852207), the Sino-German Research Group Project (No. GZ1465), and the National Key Project GJXM92579. It is a pleasure to thank C. Dafermos and M. Slemrod for many useful comments.

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Authors and Affiliations

  1. Institute of Mathematics, Hebrew University of Jerusalem, 91904, Jerusalem, Israel

    Matania Ben-Artzi

  2. Academy of Multidisciplinary Studies, Capital Normal University, Beijing, 100048, China

    Jiequan Li

  3. State Key Laboratory for Turbulence Research and Complex System, Peking University, Beijing, 100871, China

    Jiequan Li

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  1. Matania Ben-Artzi
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  2. Jiequan Li
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Correspondence to Jiequan Li.

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The manuscript is not submitted to other journals for simultaneous consideration. The submitted work is original and is not published elsewhere in any form or language. The authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest (such as honorarium; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.

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Ben-Artzi, M., Li, J. Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws. Commun. Appl. Math. Comput. 5, 1289–1298 (2023). https://doi.org/10.1007/s42967-022-00224-y

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  • DOI: https://doi.org/10.1007/s42967-022-00224-y

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