Handbook of Differential Equations: Evolutionary Equations

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ISBN: 9780444520487 | 0444520481
Cover: Hardcover
Copyright: 11/30/2005

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This book contains several introductory texts concerning the main directions in the theory of evolutionary partial differential equations. The main objective is to present clear, rigorous, and in depth surveys on the most important aspects of the present theory. The table of contents includes: W.Arendt: Semigroups and evolution equations: Calculus, regularity and kernel estimates A.Bressan: The front tracking method for systems of conservation laws E.DiBenedetto, J.M.Urbano,V.Vespri: Current issues on singular and degenerate evolution equations; L.Hsiao, S.Jiang: Nonlinear hyperbolic-parabolic coupled systems A.Lunardi: Nonlinear parabolic equations and systems D.Serre:L1-stability of nonlinear waves in scalar conservation laws B.Perthame:Kinetic formulations of parabolic and hyperbolic PDEs: from theory to numerics

Preface

List of Contributors

vii

Contents of Volume I

Euler Equations and Related Hyperbolic Conservation Laws

(104)

G.-Q. Chen

Introduction

(3)

Basic features and phenomena

(20)

Convex entropy and symmetrization

(8)

Hyperbolicity

(3)

Genuine nonlinearity

(4)

Singularities

(3)

BV bound

(2)

One-dimensional Euler equations

(11)

Isentropic Euler equations

(5)

Isothermal Euler equations

(2)

Adiabatic Euler equations

(4)

Multidimensional Euler equations and related models

(9)

The potential flow equation

(1)

Incompressible Euler equations

(1)

The transonic small disturbance equation

(1)

Pressure-gradient equations

(2)

Pressureless Euler equations

(1)

Euler equations in nonlinear elastodynamics

(1)

The Born-Infeld system in electromagnetism

(1)

Lax systems

(1)

Multidimensional steady supersonic problems

(7)

Wedge problems involving supersonic shocks

(4)

Stability of supersonic vortex sheets

(2)

Multidimensional steady transonic problems

(11)

Transonic shock problems in Rd

(2)

Nozzle problems involving transonic shocks

(3)

Free boundary approaches

(3)

Multidimensional unsteady problems

(15)

Spherically symmetric solutions

(11)

Self-similar solutions

(2)

Global solutions with special Cauchy data

(2)

Divergence-measure fields and hyperbolic conservation laws

(26)

Connections

(2)

Basic properties of divergence-measure fields

(2)

Normal traces and the Gauss-Green formula

(8)

Acknowledgments

(1)

References

(14)

Blow-up of Solutions of Supercritical Parabolic Equations

105

(54)

M. Fila

Introduction

107

(1)

Beyond blow-up

108

(2)

Global L1-solutions

108

(1)

Complete blow-up

109

(1)

Self-similar solutions

110

(4)

Backward self-similar solutions for the power

110

(2)

Forward self-similar solutions for the power

112

(1)

Backward self-similar solutions for the exponential

113

(1)

Forward self-similar solutions for the exponential

114

(1)

Examples of peaking solutions for the Cauchy problem

114

(1)

Boundedness of global solutions

115

(2)

Blow-up rate

117

(2)

Definition of type I and type II blow-up

117

(1)

Type l blow-up

117

(1)

Type II blow-up

118

(1)

Convergence to backward self-similar solutions

119

(2)

Connecting equilibria by blow-up solutions

121

(4)

Immediate regularization after blow-up

125

(16)

Statement and remarks

127

(2)

Singular stationary solutions

129

(1)

Zero number properties for singular solutions

129

(3)

Rescaled equations

132

(3)

Singular stationary solutions for the rescaled equation

135

(1)

Proof of Theorem 8.4

136

(4)

Example of a peaking solution on a ball

140

(1)

Multiple blow-up

141

(2)

Grow-up rate

143

(9)

Grow-up rate for Dirichlet problems

143

(1)

Grow-up rate for a Cauchy problem

144

(4)

Inner expansion

148

(2)

Outer expansion

150

(1)

Matching of inner and outer expansions

151

(1)

An optimal upper bound

152

(1)

Oscillating grow-up solutions and grow-up set

152

(7)

Acknowledgments

153

(1)

References

154

(5)

The Boltzmann Equation and Its Hydrodynamic Limits

159

(144)

F. Golse

Introduction

161

(2)

Fluid dynamics: A presentation of models

163

(11)

The compressible Euler system

165

(2)

The compressible Navier-Stokes system

167

(1)

The acoustic system

168

(1)

The incompressible Euler equations

169

(2)

The incompressible Navier--Stokes equations

171

(1)

The temperature equation for incompressible flows

172

(1)

Coupling of the velocity and temperature fields by conservative forces

172

(2)

The Boltzmann equation and its formal properties

174

(35)

Conservation laws

177

(4)

Boltzmann's H-theorem

181

(3)

H-theorem and a priori estimates

184

(8)

Further remarks on the H-theorem

192

(3)

The collision kernel

195

(7)

The linearized collision integral

202

(7)

Hydrodynamic scalings for the Boltzmann equation

209

(4)

Notion of a rarefied gas

209

(2)

The dimensionless Boltzmann equation

211

(2)

Compressible limits of the Boltzmann equation: Formal results

213

(13)

The compressible Euler limit: The Hilbert expansion

213

(4)

The compressible Navier-Stokes limit: The Chapman--Enskog expansion

217

(4)

The compressible Euler limit: The moment method

221

(3)

The acoustic limit

224

(2)

Incompressible limits of the Boltzmann equation: Formal results

226

(12)

The incompressible Navier-Stokes limit

226

(7)

The incompressible Stokes and Euler limits

233

(3)

Other incompressible models

236

(2)

Mathematical theory of the Cauchy problem for hydrodynamic models

238

(10)

The Stokes and acoustic systems

238

(2)

The incompressible Navier-Stokes equations

240

(2)

The incompressible Navier-Stokes-Fourier system

242

(1)

The compressible Euler system

243

(2)

The incompressible Euler equations

245

(3)

Mathematical theory of the Cauchy problem for the Boltzmann equation

248

(13)

Global classical solutions for ``small'' data

248

(1)

The DiPerna-Lions theory

249

(10)

Variants of the DiPerna-Lions theory

259

(2)

The Hilbert expansion method: Application to the compressible Euler limit

261

(4)

The relative entropy method: Application to the incompressible Euler limit

265

(4)

Applications of the moment method

269

(26)

The acoustic limit

269

(2)

The Stokes--Fourier limit

271

(2)

The Navier-Stokes-Fourier limit

273

(1)

Sketch of the proof of the Navier-Stokes-Fourier limit by the moment method

274

(15)

The nonlinear compactness estimate

289

(6)

Conclusions and open problems

295

(8)

Acknowledgments

297

(1)

References

297

(6)

Long-Time Behavior of Solutions to Hyperbolic Equations with Hysteresis

303

(68)

P. Krejci

Introduction

305

(1)

An initial-boundary value problem

306

(32)

Existence and uniqueness of solutions

308

(6)

Global boundedness

314

(8)

Asymptotic stabilization

322

(13)

Quasilinear perturbations

335

(3)

Periodic solutions

338

(14)

Statement of main results

339

(2)

Existence

341

(6)

Uniqueness

347

(2)

Asymptotic stability

349

(3)

Hysteresis operators

352

(19)

The play operator

352

(6)

Prandtl-Ishlinskii operator

358

(4)

Monotonicity

362

(1)

Energy dissipation

363

(5)

Parameter dependent hysteresis

368

(1)

Acknowledgment

369

(1)

References

369

(2)

Mathematical Issues Concerning the Navier-Stokes Equations and Some of Its Generalizations

371

(4)

J. Malek

K.R. Rajagopal

Part A. Incompressible fluids with shear, pressure and density dependent viscosity from the point of view of continuum physics

375

(24)

Introduction

375

(10)

What is a fluid?

375

(1)

Navier-Stokes fluid model

376

(1)

Departures from Newtonian behavior

377

(8)

Balance equations

385

(8)

Kinematics

385

(1)

Balance of mass -- incompressibility -- inhomogeneity

386

(2)

Balance of linear momentum

388

(1)

Balance of angular momentum

389

(1)

Balance of energy

389

(1)

Further thermodynamic considerations (the second law). Reduced dissipation equation

389

(2)

Isothermal flows at uniform temperature

391

(2)

Natural configurations

393

(1)

The constitutive models for compressible and incompressible Navier--Stokes fluids and some of their generalizations

393

(3)

Standard approach

393

(1)

Alternate approach

394

(2)

Boundary conditions

396

(3)

Part B. Mathematical analysis of flows of fluids with shear, pressure and density dependent viscosity

399

(250)

Introduction

399

(7)

A taxonomy of models

399

(2)

Mathematical self-consistency of the models

401

(1)

Weak solution: A natural notion of a solution for PDEs of the continuum physics

401

(3)

Models and their invariance with respect to scaling

404

(2)

Definitions of (suitable) weak solutions

406

(7)

Assumptions concerning the stress tensor

406

(2)

Function spaces

408

(1)

Definition of the problem (P) and its (suitable) weak solutions

408

(2)

Useful inequalities

410

(3)

Existence of a (suitable) weak solution

413

(19)

Formulation of the results and bibliographical notes

413

(1)

Definition of an approximate problem (Pε,η) and a priori estimates

414

(3)

Solvability of an approximate problem

417

(3)

Further uniform estimates w.r.t. ε and η

420

(2)

Limit ε →

422

(2)

Limit η → 0, the case r ≥ 11/5

424

(1)

Limit η → 0, the case 8/5 < r < 11/5

424

(6)

Continuity w.r.t. time in the weak topology of L2per

430

(1)

(Local) Energy equality and inequality

431

(1)

Initial conditions

431

(1)

On the smoothness of flows

432

(9)

A survey of regularity results

432

(4)

A cascade of inequalities for Ladyzhenskaya's equations

436

(2)

Boundedness of the velocity

438

(1)

Fractional higher differentiability

439

(1)

Short-time or small-data existence of a ``smooth'' solution

439

(2)

Uniqueness and large-data behavior

441

(7)

Uniquely determined flows for Ladyzhenskaya's equations

442

(3)

Long-time behavior -- the method of trajectories

445

(3)

On the structure of possible singularities for flows of the Navier-Stokes fluid

448

(3)

Other incompressible fluid models

451

(10)

Fluids with pressure dependent viscosity

451

(1)

Fluids with pressure and shear dependent viscosities

451

(1)

Inhomogeneous incompressible fluids

452

(1)

Acknowledgments

453

(1)

References

453

(8)

Evolution of Rate-Independent Systems

461

(100)

A. Mielke

Introduction

463

(5)

The simple case with a quadratic energy

468

(16)

Equivalent formulations

468

(3)

Basic a priori estimates and uniqueness

471

(2)

Basic existence theory

473

(7)

Continuity properties of the solution operator

480

(4)

Incremental problems and a priori estimates

484

(11)

Abstract setup of the problem

485

(2)

The time-incremental problem

487

(1)

Energetic a priori bounds

488

(2)

The condensed and reduced incremental problem

490

(1)

Lipschitz bounds via convexity

491

(3)

A simplified incremental problem

494

(1)

Convex energies

495

(15)

Translation invariant dissipation potentials

496

(5)

Quasi-variational inequalities

501

(3)

General dissipation metrics

504

(2)

Higher temporal regularity and improved convergence

506

(4)

Nonconvex and nonsmooth problems

510

(22)

Existence results

510

(8)

Closedness of the stable set

518

(3)

An example of nonconvergence

521

(2)

Formulations which resolve jumps

523

(2)

Time-dependent state spaces Y(t)

525

(5)

Relaxation of rate-independent systems

530

(2)

Nonassociated dissipation laws

532

(10)

General setup

532

(1)

Existence theory

533

(4)

Dry friction on surfaces

537

(5)

Applications to continuum mechanics

542

(19)

Linearized elastoplasticity

543

(1)

Finite-strain elastoplasticity

544

(3)

Phase transformations in shape-memory alloys

547

(3)

Models in ferromagnetism

550

(1)

A delamination problem

551

(2)

Crack growth in brittle materials

553

(2)

Acknowledgments

555

(1)

References

555

(6)

On the Global Weak Solutions to a Variational Wave Equation

561

(88)

P. Zhang

Y. Zheng

Introduction

563

(5)

Origin of the variational wave equation

563

(2)

Asymptotic equations

565

(1)

Young measure

566

(1)

Relation to Camassa--Holm equation

566

(1)

Open problems

567

(1)

The first asymptotic equation

568

(36)

Introduction

568

(4)

Existence of dissipative solutions

572

(11)

Uniqueness of dissipative solutions

583

(12)

Conservative solutions

595

(5)

The uniqueness of solution to the ODE

600

(4)

Rarefactive solutions to (0.0.1)

604

(26)

Introduction

604

(3)

The viscous approximation

607

(3)

Estimates

610

(2)

Classical solutions

612

(3)

Precompactness

615

(8)

Square integrable data

623

(7)

Weak solutions to (0.0.1)

630

(19)

Introduction

630

(1)

Approximate solutions and uniform estimates

631

(4)

Precompactness

635

(10)

References

645

(4)

Author Index

649

(10)

Subject Index

659

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Handbook of Differential Equations: Evolutionary Equations

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Table of Contents

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Handbook of Differential Equations: Evolutionary Equations

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