Explosive Instabilities in Mechanics

ISBN-13: 9783540635895 / Angielski / Twarda / 1998 / 197 str.

B. Straughan; Brian Straughan

Opis

This book deals with blow-up of a solution to a system of PDEs that arise in practical situations. It begins with a relatively simple account of blow-up in systems of interaction-diffusion equations, then concentrates on mechanics. 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, stressing how these equations arise in mechanics, e.g. in combustion theory. Further topics are chemotaxis in mathematical biology, change of type, from hyperbolic to elliptic, instability in soils, instability in sea ice dynamics, instability in pressure-dependent viscosity flow, and energy growth in parallel shear flows. It addresses graduate students and researchers in mechanics.

Recenzje

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)

Spis treści

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.

Blurb

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.