Explosive Instabilities in Mechanics

1. Front Matter

   Pages I-IX

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2. Introduction

   • Brian Straughan

   Pages 1-31

3. Analysis of a First-Order System

   • Brian Straughan

   Pages 33-61

4. Singularities for Classical Fluid Equations

   • Brian Straughan

   Pages 63-79

5. Catastrophic Behaviour in Other Non-linear Fluid Theories

   • Brian Straughan

   Pages 81-118

6. Blow-Up in Volterra Equations

   • Brian Straughan

   Pages 119-129

7. Chemotaxis

   • Brian Straughan

   Pages 131-142

8. Change of Type

   • Brian Straughan

   Pages 143-154

9. Rapid Energy Growth in Parallel Flows

   • Brian Straughan

   Pages 155-178

10. Back Matter

    Pages 179-197

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The subject of blow-up in a finite time, or at least very rapid growth, of a solution to a partial differential equation has been an area of intense re­ search activity in mathematics. Some ofthe early techniques and results were discussed in the monograph by Payne (1975) and in my earlier monograph, Straughan (1982). Relatively recent accounts of blow-up work in partial dif­ ferential equations may be found in the review by Levine (1990) and in the book by Samarskii et al. (1994). It is becoming increasingly clear that very rapid instabilities and, indeed, finite time blow-up are being witnessed also in problems in applied mathematics and mechanics. Also in vogue in the mathematical literature are studies of blow-up in systems of partial differen­ tial equations, partial differential equations with non-linear convection terms, and systems of partial differential equations which contain convection terms. Such equations are often derived from models of mundane situations in real life. This book is an account of these topics in a selection of areas of applied mathematics which either I have worked in or I find particularly interesting and deem relevant to be included in such an exposition. I believe the results given in Chap. 2 and Sects. 4. 2. 3 and 4. 2. 4 are new. This research was partly supported by a Max Planck Forschungspreis from the Alexander von Humboldt Foundation and the Max Planck Institute.

• dynamics
• finite element method
• finite elements
• instability
• mechanics
• stability
• stress

From the reviews
"... this book contains a clear account of exciting works in various parts of science, concentrating on blow-up solutions of systems of partial differential equations."
(J. Cugnon in: Physicalia)

• Department of Mathematics, University of Glasgow, Glasgow, Scotland

  Brian Straughan

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