### Abstract

We study the large-time asymptotic shock-front speed in an inviscid Burgers equation with a spatially random flux function. This equation is a prototype for a class of scalar conservation laws with spatial random coefficients such as the well-known Buckley-Leverett equation for two-phase flows, and the contaminant transport equation in groundwater flows. The initial condition is a shock located at the origin (the indicator function of the negative real line). We first regularize the equation by a special random viscous term so that the resulting equation can be solved explicitly by a Cole-Hopf formula. Using the invariance principle of the underlying random processes and the Laplace method, we prove that for large times the solutions behave like fronts moving at averaged constant speeds in the sense of distribution. However, the front locations are random, and we show explicitly the probability of observing the head or tail of the fronts. Finally, we pass to the inviscid limit, and establish the same results for the inviscid shock fronts.

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

Pages (from-to) | 843-871 |

Number of pages | 29 |

Journal | Journal of Statistical Physics |

Volume | 88 |

Issue number | 3-4 |

State | Published - Aug 1997 |

### Fingerprint

### Keywords

- Asymptotic speed
- Burgers equation
- Cole-Hopf formula
- Front solutions
- Random flux

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Journal of Statistical Physics*,

*88*(3-4), 843-871.

**Front speed in the Burgers equation with a random flux.** / Wehr, Jan; Xin, J.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 88, no. 3-4, pp. 843-871.

}

TY - JOUR

T1 - Front speed in the Burgers equation with a random flux

AU - Wehr, Jan

AU - Xin, J.

PY - 1997/8

Y1 - 1997/8

N2 - We study the large-time asymptotic shock-front speed in an inviscid Burgers equation with a spatially random flux function. This equation is a prototype for a class of scalar conservation laws with spatial random coefficients such as the well-known Buckley-Leverett equation for two-phase flows, and the contaminant transport equation in groundwater flows. The initial condition is a shock located at the origin (the indicator function of the negative real line). We first regularize the equation by a special random viscous term so that the resulting equation can be solved explicitly by a Cole-Hopf formula. Using the invariance principle of the underlying random processes and the Laplace method, we prove that for large times the solutions behave like fronts moving at averaged constant speeds in the sense of distribution. However, the front locations are random, and we show explicitly the probability of observing the head or tail of the fronts. Finally, we pass to the inviscid limit, and establish the same results for the inviscid shock fronts.

AB - We study the large-time asymptotic shock-front speed in an inviscid Burgers equation with a spatially random flux function. This equation is a prototype for a class of scalar conservation laws with spatial random coefficients such as the well-known Buckley-Leverett equation for two-phase flows, and the contaminant transport equation in groundwater flows. The initial condition is a shock located at the origin (the indicator function of the negative real line). We first regularize the equation by a special random viscous term so that the resulting equation can be solved explicitly by a Cole-Hopf formula. Using the invariance principle of the underlying random processes and the Laplace method, we prove that for large times the solutions behave like fronts moving at averaged constant speeds in the sense of distribution. However, the front locations are random, and we show explicitly the probability of observing the head or tail of the fronts. Finally, we pass to the inviscid limit, and establish the same results for the inviscid shock fronts.

KW - Asymptotic speed

KW - Burgers equation

KW - Cole-Hopf formula

KW - Front solutions

KW - Random flux

UR - http://www.scopus.com/inward/record.url?scp=0031206412&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031206412&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031206412

VL - 88

SP - 843

EP - 871

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -