Handbook of Differential Equations: Evolutionary Equations (eBook)

Preface 6
List of Contributors 8
Contents 10
Contents of Volume I 12
Euler Equations and Related Hyperbolic Conservation Laws 14
Introduction 16
Basic features and phenomena 19
One-dimensional Euler equations 39
Multidimensional Euler equations and related models 50
Multidimensional steady supersonic problems 59
Multidimensional steady transonic problems 66
Multidimensional unsteady problems 77
Divergence-measure fields and hyperbolic conservation laws 92
Acknowledgments 104
References 104
Blow-up of Solutions of SupercriticalParabolic Equations 118
Introduction 120
Beyond blow-up 121
Self-similar solutions 123
Examples of peaking solutions for the Cauchy problem 127
Boundedness of global solutions 128
Blow-up rate 130
Convergence to backward self-similar solutions 132
Connecting equilibria by blow-up solutions 134
Immediate regularization after blow-up 138
Multiple blow-up 154
Grow-up rate 156
Oscillating grow-up solutions and grow-up set 165
Acknowledgments 166
References 167
The Boltzmann Equation and Its Hydrodynamic Limits 172
Introduction 174
Fluid dynamics: A presentation of models 176
The Boltzmann equation and its formal properties 187
Hydrodynamic scalings for the Boltzmann equation 222
Compressible limits of the Boltzmann equation: Formal results 226
Incompressible limits of the Boltzmann equation: Formal results 239
Mathematical theory of the Cauchy problem for hydrodynamic models 251
Mathematical theory of the Cauchy problem for the Boltzmann equation 261
The Hilbert expansion method: Application to the compressible Euler limit 274
The relative entropy method: Application to the incompressible Euler limit 278
Applications of the moment method 282
Conclusions and open problems 308
Acknowledgments 310
References 310
Long-Time Behavior of Solutions toHyperbolic Equations with Hysteresis 316
Introduction 318
An initial-boundary value problem 319
Periodic solutions 351
Hysteresis operators 365
Acknowledgment 382
References 382
Mathematical Issues Concerningthe Navier-Stokes Equations andSome of Its Generalizations 384
Part A. Incompressible fluids with shear, pressure and density dependent viscosity from the point of view of continuum physics 388
Introduction 388
Balance equations 398
The constitutive models for compressible and incompressible Navier-Stokes fluids and some of their generalizations 406
Boundary conditions 409
Part B. Mathematical analysis of flows of fluids with shear, pressure and density dependent viscosity 412
Introduction 412
Definitions of (suitable) weak solutions 419
Existence of a (suitable) weak solution 426
On the smoothness of flows 445
Uniqueness and large-data behavior 454
On the structure of possible singularities for flows of the Navier-Stokes fluid 461
Other incompressible fluid models 464
Acknowledgments 466
References 466
Evolution of Rate-Independent Systems 474
Introduction 476
The simple case with a quadratic energy 481
Incremental problems and a priori estimates 497
Convex energies 508
Nonconvex and nonsmooth problems 523
Nonassociated dissipation laws 545
Applications to continuum mechanics 555
Acknowledgments 568
References 568
On the Global Weak Solutions to a Variational Wave Equation 574
Introduction 576
The first asymptotic equation 581
Rarefactive solutions to (0.0.1) 617
Weak solutions to (0.0.1) 643
References 658
Author Index 662
Subject Index 672