TY - JOUR

T1 - Vanishing viscosity solutions for conservation laws with regulated flux

AU - Bressan, Alberto

AU - Guerra, Graziano

AU - Shen, Wen

N1 - Funding Information:
The research of the first author was partially supported by NSF , with grant DMS-1411786 : Hyperbolic Conservation Laws and Applications. The research of the second author was partially supported by the PRIN 2015 project Hyperbolic Systems of Conservation Laws and Fluid Dynamics: Analysis and Applications. The second author acknowledges the hospitality of the Department of Mathematics, Penn State University – May 2017. Finally, all three authors would like to thank the anonymous referee for carefully reading the manuscript and providing many useful suggestions.
Funding Information:
The research of the first author was partially supported by NSF, with grant DMS-1411786: Hyperbolic Conservation Laws and Applications. The research of the second author was partially supported by the PRIN 2015 project Hyperbolic Systems of Conservation Laws and Fluid Dynamics: Analysis and Applications. The second author acknowledges the hospitality of the Department of Mathematics, Penn State University – May 2017. Finally, all three authors would like to thank the anonymous referee for carefully reading the manuscript and providing many useful suggestions.
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

VL - 266

SP - 312

EP - 351

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 1

ER -