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…mehr
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.
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.
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.
Rezensionen
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)
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