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Blow-up in a System of Partial Differential Equations with Conserved First Integral. Part II: Problems with Convection

Budd, C. J. and Dold, J. W. and Stuart, A. M. (1994) Blow-up in a System of Partial Differential Equations with Conserved First Integral. Part II: Problems with Convection. SIAM Journal on Applied Mathematics, 54 (3). pp. 610-640. ISSN 0036-1399. https://resolver.caltech.edu/CaltechAUTHORS:20170613-120606856

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Abstract

A reaction-diffusion-convection equation with a nonlocal term is studied; the nonlocal operator acts to conserve the spatial integral of the unknown function as time evolves. The equations are parameterised by µ, and for µ = 1 the equation arises as a similarity solution of the Navier-Stokes equations and the nonlocal term plays the role of pressure. For µ = 0, the equation is a nonlocal reaction-diffusion problem. The aim of the paper is to determine for which values of the parameter µ blow-up occurs and to study its form. In particular, interest is focused on the three cases µ < 1/2, µ > 1/2, and µ → 1. It is observed that, for any 0 ≤ µ ≤ 1/2, nonuniform global blow-up occurs; if 1/2 < µ < 1, then the blow-up is global and uniform, while for µ = 1 (the Navier-Stokes equations) there are exact solutions with initial data of arbitrarily large L_∞, L_2, and H^1 norms that decay to zero. Furthermore, one of these exact solutions is proved to be nonlinearly stable in L_2 for arbitrarily large supremum norm. An understanding of this transition from blow-up behaviour to decay behaviour is achieved by a combination of analysis, asymptotics, and numerical techniques.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1137/S0036139992232131DOIArticle
http://epubs.siam.org/doi/abs/10.1137/S0036139992232131PublisherArticle
Additional Information:© 1994 Society for Industrial and Applied Mathematics. Received by the editors June 9, 1992; accepted for publication (in revised form) July 12, 1993. We are grateful to J. T. Stuart for related discussions and for suggesting the similarity solution outlined in §2, and to V. Galktionov, E. Suli, and M. Floater for useful advice.
Subject Keywords:nonlocal source term, conserved integral, Navier-Stokes equations, exact solutions, nonlinear stability
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ25
Issue or Number:3
Classification Code:AMS subject classifications. 35B30, 35B35, 35K57, 65N50
Record Number:CaltechAUTHORS:20170613-120606856
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170613-120606856
Official Citation:Blow-up in a System of Partial Differential Equations with Conserved First Integral. Part II: Problems with Convection C. J. Budd, J. W. Dold, and A. M. Stuart SIAM Journal on Applied Mathematics 1994 54:3, 610-640
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78162
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:13 Jun 2017 19:33
Last Modified:03 Oct 2019 18:06

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