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Small dispersion approximation of shock wave dynamics

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Abstract

We introduce a dispersion approximation model for weak, entropy solutions of multidimensional scalar conservation laws using variational kinetic representation, where equilibrium densities satisfy Gibb’s entropy minimization principle for a piecewise linear, convex entropy. For such solutions, we show that small scale discontinuities, measured by the entropy increments, propagate with characteristic velocities, while the large-scale, shock-type discontinuities propagate with speeds close to the speeds of classical shock waves. In the zero-limit of the scale parameter, approximate solutions converge to a unique, entropy solution of a scalar conservation law.

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Acknowledgements

The author would like to thank an anonymous referee for proofreading the manuscript and providing valuable suggestions for improving the presentation.

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  1. Department of Mathematics, University of Houston, 4800 Calhoun Rd., Houston, TX, 77204, US

    Misha Perepelitsa

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  1. Misha Perepelitsa
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Correspondence to Misha Perepelitsa.

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Perepelitsa, M. Small dispersion approximation of shock wave dynamics. Z. Angew. Math. Phys. 73, 130 (2022). https://doi.org/10.1007/s00033-022-01713-z

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  • DOI: https://doi.org/10.1007/s00033-022-01713-z

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