Solutions to a class of one-dimensional conservation laws with discontinuous flux are constructed relying on the Crandall--Liggett theory of nonlinear contractive semigroups [H. Brézis and A. Pazy, J. Functional Analysis, 9 (1972), pp. 63--74, M. G. Crandall and T. M. Liggett, Amer. J. Math., 93 (1971), pp. 265--298], with a vanishing viscosity approach. The solutions to the corresponding viscous conservation laws are studied using the backward Euler approximations. We prove their convergence to a unique vanishing viscosity solution to the Cauchy problem for the nonviscous equations as the viscous parameter tends to zero. This approach allows us to avoid the technicalities in existing literature, such as traces, Riemann problems, interface conditions, compensated compactness and entropy inequalities. Consequently, we establish our result under very mild assumptions on the flux, with only a requirement on the smoothness with respect to the unknown variable and a condition that allows the application of the maximum principle

Guerra, G., Shen, W. (2019). Vanishing Viscosity and Backward Euler Approximations for Conservation Laws with Discontinuous Flux. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 51(4), 3112-3144 [10.1137/18M1205662].

Vanishing Viscosity and Backward Euler Approximations for Conservation Laws with Discontinuous Flux

Guerra, Graziano
;
2019

Abstract

Solutions to a class of one-dimensional conservation laws with discontinuous flux are constructed relying on the Crandall--Liggett theory of nonlinear contractive semigroups [H. Brézis and A. Pazy, J. Functional Analysis, 9 (1972), pp. 63--74, M. G. Crandall and T. M. Liggett, Amer. J. Math., 93 (1971), pp. 265--298], with a vanishing viscosity approach. The solutions to the corresponding viscous conservation laws are studied using the backward Euler approximations. We prove their convergence to a unique vanishing viscosity solution to the Cauchy problem for the nonviscous equations as the viscous parameter tends to zero. This approach allows us to avoid the technicalities in existing literature, such as traces, Riemann problems, interface conditions, compensated compactness and entropy inequalities. Consequently, we establish our result under very mild assumptions on the flux, with only a requirement on the smoothness with respect to the unknown variable and a condition that allows the application of the maximum principle
Articolo in rivista - Articolo scientifico
scalar conservation laws, discontinuous flux, vanishing viscosity, nonlinear semi-groups, backward Euler approximation
English
30-lug-2019
2019
51
4
3112
3144
partially_open
Guerra, G., Shen, W. (2019). Vanishing Viscosity and Backward Euler Approximations for Conservation Laws with Discontinuous Flux. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 51(4), 3112-3144 [10.1137/18M1205662].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/239667
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