Front speed in the Burgers equation with a random flux

Jan Wehr, J. Xin

Research output: Contribution to journalArticle

11 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)843-871
Number of pages29
JournalJournal of Statistical Physics
Volume88
Issue number3-4
StatePublished - Aug 1997

Fingerprint

Burger equation
Burgers Equation
Shock
shock fronts
Inviscid Limit
Laplace's Method
Contaminant Transport
Large Time Asymptotics
Groundwater Flow
Indicator function
Scalar Conservation Laws
Random Coefficients
Invariance Principle
random processes
two phase flow
Two-phase Flow
Random process
ground water
conservation laws
Transport Equation

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

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

In: Journal of Statistical Physics, Vol. 88, No. 3-4, 08.1997, p. 843-871.

Research output: Contribution to journalArticle

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