In this paper we introduce a concept of “regulated function” v(t,x) of two variables, which reduces to the classical definition when v is independent of t. We then consider a scalar conservation law of the form ut+F(v(t,x),u)x=0, where F is smooth and v is a regulated function, possibly discontinuous w.r.t. both t and x. By adding a small viscosity, one obtains a well posed parabolic equation. As the viscous term goes to zero, the uniqueness of the vanishing viscosity limit is proved, relying on comparison estimates for solutions to the corresponding Hamilton–Jacobi equation. As an application, we obtain the existence and uniqueness of solutions for a class of 2×2 triangular systems of conservation laws with hyperbolic degeneracy.

Bressan, A., Guerra, G., Shen, W. (2019). Vanishing viscosity solutions for conservation laws with regulated flux. JOURNAL OF DIFFERENTIAL EQUATIONS, 266(1), 312-351 [10.1016/j.jde.2018.07.044].

Vanishing viscosity solutions for conservation laws with regulated flux

Guerra, G
;
2019

Abstract

In this paper we introduce a concept of “regulated function” v(t,x) of two variables, which reduces to the classical definition when v is independent of t. We then consider a scalar conservation law of the form ut+F(v(t,x),u)x=0, where F is smooth and v is a regulated function, possibly discontinuous w.r.t. both t and x. By adding a small viscosity, one obtains a well posed parabolic equation. As the viscous term goes to zero, the uniqueness of the vanishing viscosity limit is proved, relying on comparison estimates for solutions to the corresponding Hamilton–Jacobi equation. As an application, we obtain the existence and uniqueness of solutions for a class of 2×2 triangular systems of conservation laws with hyperbolic degeneracy.
Articolo in rivista - Articolo scientifico
Conservation law with discontinuous flux; Existence and uniqueness; Hamilton–Jacobi equation; Regulated flux function; Vanishing viscosity; Analysis; Applied Mathematics
English
2019
2019
266
1
312
351
reserved
Bressan, A., Guerra, G., Shen, W. (2019). Vanishing viscosity solutions for conservation laws with regulated flux. JOURNAL OF DIFFERENTIAL EQUATIONS, 266(1), 312-351 [10.1016/j.jde.2018.07.044].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/213887
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