Connections between the convective diffusion equation and the forced Burgers equation
type: Item
instance of: scholarly article
title: Connections between the convective diffusion equation and the forced Burgers equation
zbMATH Open document ID: 1043.35008
DOI: 10.1155/S1048953302000060
author: Nejib Smaoui
author: F. B. M. Belgacem
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 2002-08-08T00:00:00Z
review text: Summary: The convective diffusion equation with drift \(b(x)\) and indefinite weight \(r(x)\), \[ \frac{\partial\varphi} {\partial t}= \frac{\partial}{\partial x}\left[a\frac{\partial\varphi}{\partial x}-b(x) \varphi\right] +\lambda r(x)\varphi, \tag{1} \] is introduced as a model for population dispersal. Strong connections between equation (1) and the forced Burgers equation with positive frequency \((m\geq 0)\), \[ \frac {\partial u}{\partial t}= \frac{\partial^2u} {\partial x^2}-u \frac {\partial u}{\partial x}+ mu+k(x), \tag{2} \] are established through the Hopf-Cole transformation. Equation (2) is a prime prototype of the large class of quasi-linear parabolic equations given by \[ \frac{\partial v} {\partial t}=\frac {\partial^2v}{\partial x^2}+ \frac{\partial\bigl(f(v) \bigr)} {\partial x}+g(v)+h(x). \tag{3} \] A compact attractor and an inertial manifold for the forced Burgers equation are shown to exist via the Kwak transformation. Consequently, existence of an inertial manifold for the convective diffusion equation is guaranteed.
Mathematics Subject Classification ID: 35A22
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37L25
Mathematics Subject Classification ID: 35Q80
Mathematics Subject Classification ID: 37L30
zbMATH DE Number: 1777334
zbMATH Keywords: convective diffusion equation with drift
zbMATH Keywords: indefinite weight
zbMATH Keywords: population dispersal
zbMATH Keywords: Hopf-Cole transformation
zbMATH Keywords: compact attractor
zbMATH Keywords: inertial manifold
zbMATH Keywords: forced Burgers equation
zbMATH Keywords: Kwak transformation
MaRDI profile type: Publication
full work available at URL: s1048953302000060
OpenAlex ID: W2141943741
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description: scientific article; zbMATH DE number 1777334