TY - JOUR

T1 - Vanishing viscosity solutions for conservation laws with regulated flux

AU - Bressan, Alberto

AU - Guerra, Graziano

AU - Shen, Wen

N1 - Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2019/1/5

Y1 - 2019/1/5

N2 - 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.

AB - 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.

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U2 - 10.1016/j.jde.2018.07.044

DO - 10.1016/j.jde.2018.07.044

M3 - Article

AN - SCOPUS:85054455737

SN - 0022-0396

VL - 266

SP - 312

EP - 351

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 1

ER -