Handbook of Differential Equations: Evolutionary Equations

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ISBN: 9780444528483 | 0444528482
Cover: Hardcover
Copyright: 12/11/2006

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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

Contents of Volume II

xiii

1. Gradient Flows of Probability Measures

L. Ambrosio and G. Savaré

Introduction

Notation

1. Notation and measure-theoretic results

1.1. Transport maps and transport plans

1.2. Narrow convergence

1.3. The change of variables formula

2. Metric and differentiable structure of the Wasserstein space

2.1. Absolutely continuous maps and metric derivative

2.2. The quadratic optimal transport problem

2.3. Geodesics in P2(Rd)

2.4. Existence of optimal transport maps

2.5. The continuity equation with locally Lipschitz velocity fields

2.6. The tangent bundle to the Wasserstein space

3. Convex functionals in P2 (Rd)

3.1. λ-geodesically convex functionals in P2(Rd)

3.2. Examples of convex functionals in P2(Rd)

3.3. Relative entropy and convex functionals of measures

3.4. Log-concavity and displacement convexity

4. Subdifferential calculus in P2 (Rd)

4.1. Definition of the subdifferential for a.c. measures

4.2. Subdifferential calculus in P2a(Rd)

4.3. The case of λ-convex functionals along geodesics

4.4. Regular functionals

4.5. Examples of subdifferentials

5. Gradient flows of λ-geodesically convex functionals in P2 (Rd)

5.1. Characterizations of gradient flows, uniqueness and contractivity

5.2. Main properties of gradient flows

5.3. Existence of gradient flows by convergence of the "minimizing movement" scheme

5.4. Bibliographical notes

104

6. Applications to evolution PDEs

107

6.1. Gradient flows and evolutionary PDEs of diffusion type

107

6.2. The linear transport equation for λ-convex potentials

111

6.3. Kolmogorov-Fokker-Planck equation

113

6.4. Nonlinear diffusion equations

129

6.5. Drift diffusion equations with nonlocal terms

132

6.6. Gradient flow of - W²/2 and geodesics

133

References

133

2. The Mathematics of Chemotaxis

137

M.A. Herrero

1. Introduction: What is chemotaxis?

139

2. How do chemotactic units work?

141

2.1. Ligand binding to receptors

143

2.2. The role of fluctuations

144

2.3. Diffusion effects on ligand binding

147

2.4. Estimating the measurement error

149

2.5. Receptor clustering

154

2.6. Signaling pathways and their performance properties

157

3. Some mathematical problems arising from the study of Dictyostelium discoideum

168

3.1. The social life of Dictyostelium discoideum in a nutshell

169

3.2. Early aggregation stages: the Keller—Segel model

170

3.3. The Keller—Segel model revisited: from micro to macro

175

3.4. Pattern formation in Dictyostelium discoideum

180

Acknowledgements

189

References

189

3. Examples of Singular Limits in Hydrodynamics

195

N. Masmoudi

1. Introduction

197

1.1. Dimensionless parameters

197

1.2. Mathematical problems

198

1.3. The compressible Navier—Stokes system

198

1.4. Dimensionless parameters

199

2. The inviscid limit

201

2.1. The whole space case

202

2.2. The case of the Dirichlet boundary condition

205

2.3. Weak limit

212

3. Compressible—incompressible limit

215

3.1. Formal limit

216

3.2. The case of strong solutions

217

3.3. The case of global weak solutions

223

3.4. Study of the limit γ -> infinity

233

3.5. The nonisentropic case

236

4. Study of rotating fluids at high frequency

237

4.1. The periodic case

238

4.2. Ekman boundary layers in Ω = T² x ]0, h[

238

4.3. The case of other geometries

244

4.4. Other related problems

244

5. Hydrodynamic limit of the Boltzmann equation

245

5.1. Scalings and formal asymptotics

245

5.2. The convergence toward the incompressible Navier—Stokes—Fourier system

251

5.3. The convergence toward the Stokes system

254

5.4. The case of a bounded domain

256

5.5. Convergence toward the Euler system

260

6. Some homogenization problems

261

6.1. Darcy law

264

6.2. Homogenization of a compressible model

264

6.3. Homogenization of the Euler system

266

7. Conclusion

268

7.1. Other limits

268

7.2. Concluding remarks

268

Acknowledgement

269

References

269

4. Notes on Hyperbolic Systems of Conservation Laws and Transport Equations

277

C. De Lellis

1. Introduction

279

1.1. The Keyfitz and Kranzer system

279

1.2. Bressan's compactness conjecture

280

1.3. Ambrosio's renormalization theorem

281

1.4. Well-posedness for the Keyfitz and Kranzer system

282

1.5. Renormalization conjecture for nearly incompressible BV fields

283

1.6. Plan of the paper

284

2. Preliminaries

284

2.1. Notation

284

2.2. Measure theory

285

2.3. Approximate continuity and approximate jumps

287

2.4. BV functions

288

2.5. Caccioppoli sets and coarea formula

289

2.6. The Vol'pert chain rule

289

2.7. Alberti's rank-one theorem

290

3. DiPerna–Lions theory for nearly incompressible flows

290

3.1. Lagrangian flows

291

3.2. Nearly incompressible fields and fields with the renormalization property

294

3.3. Existence and uniqueness of solutions to transport equations

298

3.4. Stability of solutions to transport equations

302

3.5. Existence, uniqueness and stability of regular Lagrangian flows

304

4. Commutator estimates and Ambrosio's renormalization theorem

308

4.1. Difference quotients of BV functions

310

4.2. Commutator estimate

312

4.3. Bouchut's lemma and Alberti's lemma

317

4.4. Proof of Theorem 4.1

320

5. Existence, uniqueness, and stability for the Keyfitz and Kranzer system

321

5.1. Proof of Theorem 5.5

324

5.2. Renormalized entropy solutions are entropy solutions

326

5.3. Proof of Proposition 5.9

330

6. Blow-up of the BV norm for the Keyfitz and Kranzer system

334

6.1. Preliminary lemmas

335

6.2. Proof of Theorem 6.1

338

6.3. Proof of Proposition 6.5

344

7. Partial regularity and trace properties of solutions to transport equations

349

7.1. Anzellotti's weak trace for measure – divergence bounded vector fields

350

7.2. Further properties of Anzellotti's weak trace

354

7.3. Change of variables for traces

357

7.4. Proof of Theorem 7.1

359

7.5. Proof of Theorem 7.4

363

8. Bressan's compactness conjecture and the renormalization conjecture for nearly incompressible BV vector fields

364

8.1. Absolutely continuous and jump parts of the measure D·(ph(w)B)

366

8.2. Proof of Proposition 8.4 and concentration of commutators

368

8.3. Proof of Theorem 8.10

369

9. Tangential sets of BV vector fields

374

Acknowledgements

380

References

380

5. Collisionless Kinetic Equations from Astrophysics — The Vlasov—Poisson System

383

G. Rein

Introduction

385

Notation and preliminaries

387

1. Classical solutions to the initial value problem

390

1.1. The initial value problem — an overview

390

1.2. Local existence

393

1.3. Spherically symmetric solutions

402

1.4. Small data solutions

408

1.5. Conservation laws and a priori bounds

413

1.6. Global existence for general data

417

1.7. Asymptotic behavior

431

2. Stability

435

2.1. Introduction — steady states, stability and energy-Casimir functionals

435

2.2. Reduction

443

2.3. Existence of minimizers via the reduced problem

449

2.4. Existence of minimizers — the direct approach

455

2.5. Minimizers are steady states

460

2.6. Dynamical stability

465

2.7. The reduced variational problem and the Euler—Poisson system

470

References

472

6. Stochastic Representations for Nonlinear Parabolic PDEs

477

H.M. Soner

1. Introduction

479

Notation

480

2. Linear case: Feynman–Kac representation

480

2.1. Linear monotone semigroups

481

2.2. Zeroth-order term and forcing

484

2.3. Diffusions and parabolic PDEs

485

2.4. Initial value problems

487

2.5. Discrete Markov processes and simply coupled equations

488

2.6. Jump Markov processes and integro-differential equations

490

2.7. Dirichlet boundary conditions

492

2.8. Neumann condition and the Skorokhod problem

493

2.9. Stationary problems

494

3. Representation via controlled processes

495

3.1. Viscosity solutions

498

3.2. Optimal control of diffusion processes

502

3.3. Smooth value function and verification

507

3.4. Optimal control of jump Markov processes

509

3.5. Other type of control problems

510

3.6. Stochastic target and geometric problems

511

4. Backward representations

514

4.1. Backward stochastic differential equations

514

4.2. Second-order backward stochastic equations

515

5. Monte Carlo methods

521

5.1. The linear case

521

5.2. The semilinear case

522

5.3. The fully nonlinear case

523

Acknowledgement

524

References

524

7. Controllability and Observability of Partial Differential Equations: Some Results and Open Problems

527

E. Zuazua

1. Introduction

529

2. Preliminaries on finite-dimensional systems

531

2.1. Problem formulation

531

2.2. Controllability identical to observability

532

2.3. Bang-bang controls

538

3. Controllability of the linear wave equation

540

3.1. Statement of the problem

540

3.2. Exact controllability

543

3.3. Ingham inequalities and Fourier series techniques

550

3.4. Approximate controllability

553

3.5. Quasibang-bang controls

556

3.6. Stabilization

559

4. The heat equation

562

4.1. Problem formulation

562

4.2. Approximate controllability

564

4.3. Null controllability

568

4.4. Parabolic equations of fractional order

572

4.5. Carleman inequalities for heat equations with potentials

574

4.6. Bang-bang controls

574

4.7. Discussion and comparison

577

5. Sharp observability estimates for the linear heat equation

578

5.1. Sharp estimates

578

5.2. Optimality

583

6. Parabolic equations with low regularity coefficients

587

6.1. Elliptic equations

588

6.2. Parabolic equations

592

Author Index

623

Subject Index

633

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