### Abstract

In the current paper, we consider large time behavior of solutions to scalar conservation laws with an artificial heat flux term. In the case where the heat flux is governed by Fourier's law, the equation is scalar viscous conservation laws. In this case, existence and asymptotic stability of one-dimensional viscous shock waves have been studied in several papers. The main concern in the current paper is a 2 × 2 system of hyperbolic equations with relaxation which is derived by prescribing Cattaneo's law for the heat flux. We consider the one-dimensional Cauchy problem for the system of Cattaneo-type and show existence and asymptotic stability of viscous shock waves. We also obtain the convergence rate by utilizing the weighted energy method. By letting the relaxation time zero in the system of Cattaneo-type, the system is formally deduced to scalar viscous conservation laws of Fourier-type. This is a singular limit problem which occurs an initial layer. We also consider the singular limit problem associated with viscous shock waves.

Original language | English |
---|---|

Pages (from-to) | 795-819 |

Number of pages | 25 |

Journal | Kinetic and Related Models |

Volume | 11 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2018 Jan 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Initial layer
- Relaxation limit
- Scalar viscous conservation laws
- Traveling waves
- Weighted energy method

### ASJC Scopus subject areas

- Numerical Analysis
- Modelling and Simulation

### Cite this

*Kinetic and Related Models*,

*11*(4), 795-819. https://doi.org/10.3934/krm.2018032

**Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law.** / Nakamura, Tohru; Kawashima, Shuichi.

Research output: Contribution to journal › Article

*Kinetic and Related Models*, vol. 11, no. 4, pp. 795-819. https://doi.org/10.3934/krm.2018032

}

TY - JOUR

T1 - Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law

AU - Nakamura, Tohru

AU - Kawashima, Shuichi

PY - 2018/1/1

Y1 - 2018/1/1

N2 - In the current paper, we consider large time behavior of solutions to scalar conservation laws with an artificial heat flux term. In the case where the heat flux is governed by Fourier's law, the equation is scalar viscous conservation laws. In this case, existence and asymptotic stability of one-dimensional viscous shock waves have been studied in several papers. The main concern in the current paper is a 2 × 2 system of hyperbolic equations with relaxation which is derived by prescribing Cattaneo's law for the heat flux. We consider the one-dimensional Cauchy problem for the system of Cattaneo-type and show existence and asymptotic stability of viscous shock waves. We also obtain the convergence rate by utilizing the weighted energy method. By letting the relaxation time zero in the system of Cattaneo-type, the system is formally deduced to scalar viscous conservation laws of Fourier-type. This is a singular limit problem which occurs an initial layer. We also consider the singular limit problem associated with viscous shock waves.

AB - In the current paper, we consider large time behavior of solutions to scalar conservation laws with an artificial heat flux term. In the case where the heat flux is governed by Fourier's law, the equation is scalar viscous conservation laws. In this case, existence and asymptotic stability of one-dimensional viscous shock waves have been studied in several papers. The main concern in the current paper is a 2 × 2 system of hyperbolic equations with relaxation which is derived by prescribing Cattaneo's law for the heat flux. We consider the one-dimensional Cauchy problem for the system of Cattaneo-type and show existence and asymptotic stability of viscous shock waves. We also obtain the convergence rate by utilizing the weighted energy method. By letting the relaxation time zero in the system of Cattaneo-type, the system is formally deduced to scalar viscous conservation laws of Fourier-type. This is a singular limit problem which occurs an initial layer. We also consider the singular limit problem associated with viscous shock waves.

KW - Initial layer

KW - Relaxation limit

KW - Scalar viscous conservation laws

KW - Traveling waves

KW - Weighted energy method

UR - http://www.scopus.com/inward/record.url?scp=85045321633&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85045321633&partnerID=8YFLogxK

U2 - 10.3934/krm.2018032

DO - 10.3934/krm.2018032

M3 - Article

VL - 11

SP - 795

EP - 819

JO - Kinetic and Related Models

JF - Kinetic and Related Models

SN - 1937-5093

IS - 4

ER -