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.
| 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
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Vanishing viscosity solutions for conservation laws with regulated flux. / Bressan, Alberto; Guerra, Graziano; Shen, Wen.
In: Journal of Differential Equations, Vol. 266, No. 1, 05.01.2019, p. 312-351.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 -