Abstract
We study a nonlocal regularisation of a scalar conservation law given by a fractional derivative of order between one and two. The nonlocal operator is of Riesz–Feller type with skewness two minus its order. This equation describes the internal structure of hydraulic jumps in a shallow water model. The main purpose of the paper is the study of the vanishing viscosity limit of the Cauchy problem for this equation. First, we study the properties of the solution of the regularised problem and then we show that the difference between the regularised solution and the entropy solution of the scalar conservation law converges to zero in this limit in \(C([0,T];L^1_{loc}({\mathbb {R}}))\) for initial data in \(L^\infty ({\mathbb {R}})\), and in \(C([0,T];L^1({\mathbb {R}}))\) for initial data in \( L^\infty ({\mathbb {R}})\cap BV({\mathbb {R}})\). In order to prove these results we use weak entropy inequalities and the double scale technique of Kruzhkov. Such techniques also allow to show the \(L^1({\mathbb {R}})\) contraction of the regularised problem. For completeness, we study the behaviour in the tail of travelling wave solutions for genuinely nonlinear fluxes. These waves converge to shock waves in the vanishing viscosity limit, but decay algebraically as \(x-ct \rightarrow \infty \), rather than exponentially, the latter being a behaviour that they exhibit as \(x-ct \rightarrow - \infty \), however. Finally, we generalise the results concerning the vanishing viscosity limit to Riesz–Feller operators.
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Acknowledgements
The authors greatly thank Franz Achleitner for helpful comments and a thorough review of the first version of the manuscript. The authors also acknowledge the financial support of the Spanish Government through the MICINNU Projects MTM2014-53145-P and PGC2018-094522-B-I00, and of the Basque Government through the Research Group Grants IT641-13 and IT1247-19. Xuban Diez-Izagirre also acknowledges the support of the Basque Government through the doctoral Grant PRE-2018-2-0013.
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Communicated by Ansgar Jüngel.
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Diez-Izagirre, X., Cuesta, C.M. Vanishing viscosity limit of a conservation law regularised by a Riesz–Feller operator. Monatsh Math 192, 513–550 (2020). https://doi.org/10.1007/s00605-020-01413-8
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DOI: https://doi.org/10.1007/s00605-020-01413-8