### 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 u_{t}+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.

Original language | English (US) |
---|---|

Pages (from-to) | 312-351 |

Number of pages | 40 |

Journal | Journal of Differential Equations |

Volume | 266 |

Issue number | 1 |

DOIs | |

State | Published - Jan 5 2019 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Journal of Differential Equations*,

*266*(1), 312-351. https://doi.org/10.1016/j.jde.2018.07.044

}

*Journal of Differential Equations*, vol. 266, no. 1, pp. 312-351. https://doi.org/10.1016/j.jde.2018.07.044

**Vanishing viscosity solutions for conservation laws with regulated flux.** / Bressan, Alberto; Guerra, Graziano; Shen, Wen.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Vanishing viscosity solutions for conservation laws with regulated flux

AU - Bressan, Alberto

AU - Guerra, Graziano

AU - Shen, Wen

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.

UR - http://www.scopus.com/inward/record.url?scp=85054455737&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85054455737&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2018.07.044

DO - 10.1016/j.jde.2018.07.044

M3 - Article

VL - 266

SP - 312

EP - 351

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 1

ER -