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Published June 1994 | Published
Journal Article Open

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


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.

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.

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August 20, 2023
August 20, 2023