### Abstract

This paper contains a qualitative study of a scalar conservation law with viscosity: u_{t} + f(u)_{x} = u_{xx}. We consider the problem of identifying the location of viscous shocks, thus obtaining an optimal finite dimensional description of solutions to the viscous conservation law. We introduce a nonlinear functional whose minimizers yield the viscous traveling profiles which optimally fit the given solution. We prove that outside an initial time interval and away from times of shock interactions, our functional remains very small, i.e., the solution can be accurately represented by a finite number of viscous traveling waves.

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

Pages (from-to) | 1474-1488 |

Number of pages | 15 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 38 |

Issue number | 5 |

DOIs | |

State | Published - Dec 1 2006 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Mathematical Analysis*,

*38*(5), 1474-1488. https://doi.org/10.1137/050642642

}

*SIAM Journal on Mathematical Analysis*, vol. 38, no. 5, pp. 1474-1488. https://doi.org/10.1137/050642642

**Optimal tracing of viscous shocks in solutions of viscous conservation laws.** / Shen, Wen; Mee, Rea Park.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Optimal tracing of viscous shocks in solutions of viscous conservation laws

AU - Shen, Wen

AU - Mee, Rea Park

PY - 2006/12/1

Y1 - 2006/12/1

N2 - This paper contains a qualitative study of a scalar conservation law with viscosity: ut + f(u)x = uxx. We consider the problem of identifying the location of viscous shocks, thus obtaining an optimal finite dimensional description of solutions to the viscous conservation law. We introduce a nonlinear functional whose minimizers yield the viscous traveling profiles which optimally fit the given solution. We prove that outside an initial time interval and away from times of shock interactions, our functional remains very small, i.e., the solution can be accurately represented by a finite number of viscous traveling waves.

AB - This paper contains a qualitative study of a scalar conservation law with viscosity: ut + f(u)x = uxx. We consider the problem of identifying the location of viscous shocks, thus obtaining an optimal finite dimensional description of solutions to the viscous conservation law. We introduce a nonlinear functional whose minimizers yield the viscous traveling profiles which optimally fit the given solution. We prove that outside an initial time interval and away from times of shock interactions, our functional remains very small, i.e., the solution can be accurately represented by a finite number of viscous traveling waves.

UR - http://www.scopus.com/inward/record.url?scp=34648827554&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34648827554&partnerID=8YFLogxK

U2 - 10.1137/050642642

DO - 10.1137/050642642

M3 - Article

AN - SCOPUS:34648827554

VL - 38

SP - 1474

EP - 1488

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 5

ER -