Explosive Instabilities in Mechanics

Explosive Instabilities in Mechanics

by Brian Straughan

Paperback(Softcover reprint of the original 1st ed. 1998)

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Overview

This book deals with blow-up, or at least very rapid growth, of a solution to a system of partial differential equations that arise in practical physics situations. It begins with a relatively simple account of blow-up in systems of interaction-diffusion equations.
Then the book concentrates on mechanics applications. In particular it deals with the Euler equations, Navier--Stokes equations, models for glacier physics, Korteweg--de-Vries equations, and ferro-hydrodynamics. Blow-up is treated in Volterra equations, too, stressing how these equations arise in mechanics, e.g. in combustion theory. The novel topic of chemotaxis in mathematical biology is also presented. There is a chapter on change of type, from hyperbolic to elliptic, addressing three new and important applications: instability in soils, instability in sea ice dynamics, and also instability in pressure-dependent viscosity flow. Finally, the book includes an exposition of exciting work, very recent and on-going, dealing with rapid energy growth in parallel shear flows.
The book addresses graduate students as well as researchers in mechanics and applied mathematics.

Product Details

ISBN-13: 9783642637407
Publisher: Springer Berlin Heidelberg
Publication date: 10/28/2012
Edition description: Softcover reprint of the original 1st ed. 1998
Pages: 197
Product dimensions: 6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

1. Introduction.- 1.1 Blow-Up in Partial Differential Equations in Applied Mathematics.- 1.2 Methods of Establishing Non-existence and Growth Solutions.- 1.2.1 The Concavity Method.- 1.2.2 The Eigenfunction Method.- 1.2.3 Explicit Inequality Methods.- 1.2.4 The Multi-Eigenfunction Method.- 1.2.5 Logarithmic Convexity.- 1.3 Finite Time Blow-Up Systems with Convection.- 1.3.1 Fujita-Type Problems.- 1.3.2 Equations with Gradient Terms.- 1.3.3 Systems with Gradient Terms.- 1.3.4 Equations with Gradient Terms and Non-Dirichlet Boundary Conditions.- 1.3.5 Blow-Up of Derivatives.- 2. Analysis of a First-Order System.- 2.1 Conditional Decay of Solutions.- 2.2 Boundedness of Solutions.- 2.3 Unconditional Decay of Solutions.- 2.3.1 Special Cases.- 2.4 Global Non-existence of Solutions.- 2.5 Numerical Results by Finite Elements.- 2.5.1 Solution Structure with Linear and Quadratic Right-Hand Sides.- 3. Singularities for Classical Fluid Equations.- 3.1 Breakdown for First-Order Systems.- 3.2 Blow-Up of Solutions to the Euler Equations.- 3.2.1 Vortex Sheet Breakdown and Rayleigh-Taylor Instability.- 3.2.2 A Mathematical Theory for Sonoluminescence.- 3.3 Blow-Up of Solutions to the Navier-Stokes Equations.- 3.3.1 Self-similar Solutions.- 3.3.2 Bénard-Marangoni Convection.- 4. Catastrophic Behaviour in Other Non-linear Fluid Theories.- 4.1 Non-existence on Unbounded Domains.- 4.1.1 Ladyzhenskaya’s Models.- 4.1.2 Global Non-existence Backward in Time for Model I, When the Spatial Domain Is R2.- 4.1.3 Global Non-existence Backward in Time for Model I, When the Spatial Domain Is R3.- 4.1.4 Exponential Growth for Model II, Backward in Time.- 4.1.5 The Backward in Time Problem for Model III.- 4.2 A Model for a Second Grade Fluid in Glacier Physics.- 4.2.1 Non-existence Forward in Time for Model I.- 4.2.2 Non-existence Backward in Time for Model I.- 4.2.3 Exponential Growth Forward in Time for Model II.- 4.2.4 Exponential Boundedness Backward in Time for Model II.- 4.3 Blow-Up for Generalised KdeV Equations.- 4.4 Very Rapid Growth in Ferrohydrodynamics.- 4.5 Temperature Blow-Up in an Ice Sheet.- 5. Blow-Up in Volterra Equations.- 5.1 Blow-Up for a Solution to a Volterra Equation.- 5.1.1 A General Non-linear Volterra Equation.- 5.1.2 Volterra Equations Motivated by Partial Differential Equations on a Bounded Spatial Domain.- 5.2 Blow-Up for a Solution to a System of Volterra Equations.- 5.2.1 Coupled Non-linear Volterra Equations Which May Arise from Non-linear Parabolic Systems.- 6. Chemotaxis.- 6.1 Mathematical Theories of Chemotaxis.- 6.1.1 A Simplified Model.- 6.2 Blow-Up in Chemotaxis When There Are Two Diffusion Terms.- 6.3 Blow-Up in Chemotaxis with a Single Diffusion Term.- 7. Change of Type.- 7.1 Instability in a Hypoplastic Material.- 7.2 Instability in a Viscous Plastic Model for Sea Ice Dynamics.- 7.3 Pressure Dependent Viscosity Flow.- 8. Rapid Energy Growth in Parallel Flows.- 8.1 Rapid Growth in Incompressible Viscous Flows.- 8.1.1 Parallel Flows.- 8.1.2 Energy Growth in Circular Pipe Flow.- 8.1.3 Linear Instability of Elliptic Pipe Flow.- 8.2 Transient Growth in Compressible Flows.- 8.3 Shear Flow in Granular Materials.- 8.4 Energy Growth in Parallel Flows of Superimposed Viscous Fluids.

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