### Abstract

We consider a special 2 x 2 viscous hyperbolic system of conservation laws of the form u_{t} + A(u)u_{x} = εu_{xx}, where A(u) = Df(u) is the Jacobian of a flux function f. For initial data with small total variation, we prove that the solutions satisfy a uniform BV bound, independent of ε. Letting ε → 0, we show that solutions of the viscous system converge to the unique entropy weak solutions of the hyperbolic system u_{t} + f(u)_{x} = 0. Within the proof, we introduce two new Lyapunov functionals which control the interaction of viscous waves of the same family. This provides a first example where uniform BV bounds and convergence of vanishing viscosity solutions are obtained, for a system with a genuinely nonlinear field where shock and rarefaction curves do not coincide.

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

Pages (from-to) | 449-476 |

Number of pages | 28 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 7 |

Issue number | 3 |

State | Published - Jul 2001 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Analysis
- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete and Continuous Dynamical Systems*,

*7*(3), 449-476.

}

*Discrete and Continuous Dynamical Systems*, vol. 7, no. 3, pp. 449-476.

**A Case study in vanishing viscosity.** / Bianchini, Stefano; Bressan, Alberto.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A Case study in vanishing viscosity

AU - Bianchini, Stefano

AU - Bressan, Alberto

PY - 2001/7

Y1 - 2001/7

N2 - We consider a special 2 x 2 viscous hyperbolic system of conservation laws of the form ut + A(u)ux = εuxx, where A(u) = Df(u) is the Jacobian of a flux function f. For initial data with small total variation, we prove that the solutions satisfy a uniform BV bound, independent of ε. Letting ε → 0, we show that solutions of the viscous system converge to the unique entropy weak solutions of the hyperbolic system ut + f(u)x = 0. Within the proof, we introduce two new Lyapunov functionals which control the interaction of viscous waves of the same family. This provides a first example where uniform BV bounds and convergence of vanishing viscosity solutions are obtained, for a system with a genuinely nonlinear field where shock and rarefaction curves do not coincide.

AB - We consider a special 2 x 2 viscous hyperbolic system of conservation laws of the form ut + A(u)ux = εuxx, where A(u) = Df(u) is the Jacobian of a flux function f. For initial data with small total variation, we prove that the solutions satisfy a uniform BV bound, independent of ε. Letting ε → 0, we show that solutions of the viscous system converge to the unique entropy weak solutions of the hyperbolic system ut + f(u)x = 0. Within the proof, we introduce two new Lyapunov functionals which control the interaction of viscous waves of the same family. This provides a first example where uniform BV bounds and convergence of vanishing viscosity solutions are obtained, for a system with a genuinely nonlinear field where shock and rarefaction curves do not coincide.

UR - http://www.scopus.com/inward/record.url?scp=0035625301&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035625301&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0035625301

VL - 7

SP - 449

EP - 476

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 3

ER -