Author	Straughan, Brian. author
Title	Explosive Instabilities in Mechanics [electronic resource] / by Brian Straughan
Imprint	Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1998
Connect to	http://dx.doi.org/10.1007/978-3-642-58807-5
Descript	IX, 197 p. online resource

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

1. Introduction -- 1.1 Blow-Up in Partial Differential Equations in Applied Mathematics -- 1.2 Methods of Establishing Non-existence and Growth Solutions -- 1.3 Finite Time Blow-Up Systems with Convection -- 2. Analysis of a First-Order System -- 2.1 Conditional Decay of Solutions -- 2.2 Boundedness of Solutions -- 2.3 Unconditional Decay of Solutions -- 2.4 Global Non-existence of Solutions -- 2.5 Numerical Results by Finite Elements -- 3. Singularities for Classical Fluid Equations -- 3.1 Breakdown for First-Order Systems -- 3.2 Blow-Up of Solutions to the Euler Equations -- 3.3 Blow-Up of Solutions to the Navier-Stokes Equations -- 4. Catastrophic Behaviour in Other Non-linear Fluid Theories -- 4.1 Non-existence on Unbounded Domains -- 4.2 A Model for a Second Grade Fluid in Glacier Physics -- 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 -- 5.2 Blow-Up for a Solution to a System of Volterra Equations -- 6. Chemotaxis -- 6.1 Mathematical Theories of Chemotaxis -- 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 -- 8.2 Transient Growth in Compressible Flows -- 8.3 Shear Flow in Granular Materials -- 8.4 Energy Growth in Parallel Flows of Superimposed Viscous Fluids

1. Physics
2. Mechanics
3. Mechanics
4. Applied
5. Physics
6. Mechanics
7. Theoretical and Applied Mechanics