Hyperbolic Conservation Laws in Continuum Physics

ISBN-13: 9783662494493 / Angielski / Twarda / 2016 / 826 str.

Constantine M. Dafermos

Opis

This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of
(a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics;
(b) specialists in continuum mechanics who may need analytical tools;
(c) experts in numerical analysis who wish to learn the underlying mathematical theory; and
(d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws.This new edition places increased emphasis on hyperbolic systems of balance laws with dissipative source, modeling relaxation phenomena. It also presents an account of recent developments on the Euler equations of compressible gas dynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised, expanded and brought up to date, and has been enriched with new applications to elasticity and differential geometry. The bibliography, also expanded and updated, now comprises close to two thousand titles.From the reviews of the 3rd edition: "This is the third edition of the famous book by C.M. Dafermos. His masterly written book is, surely, the most complete exposition in the subject." Evgeniy Panov, Zentralblatt MATH "A monumental book encompassing all aspects of the mathematical theory of hyperbolic conservation laws, widely recognized as the "Bible" on the subject." Philippe G. LeFloch, Math. Reviews

Recenzje

"This book is a contribution to the fields of mathematical analysis and partial differential equations and their broad interactions with various branches of applied mathematics and continuum physics, whose value cannot be overstated. ... a useful resource for a variety of courses as well as an almost encyclopedic reference point for advanced researchers. ... It is a great achievement that will influence the PDEs and mathematical analysis community for years to come." (Marta Lewicka, Mathematical Reviews, November, 2017)

Spis treści

I Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Formulation of the Balance Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Reduction to Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Change of Coordinates and a Trace Theorem . . . . . . . . . . . . . . . . 7

1.4 Systems of Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Companion Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Weak and Shock Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.7 Survey of the Theory of BV Functions . . . . . . . . . . . . . . . . . . . . . . 17

1.8 BV Solutions of Systems of Balance Laws . . . . . . . . . . . . . . . . . . 21

1.9 Rapid Oscillations and the Stabilizing Effect of Companion

Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

II Introduction to Continuum Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Balance Laws in Continuum Physics . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 The Balance Laws of Continuum Thermomechanics . . . . . . . . . . 31

2.4 Material Frame Indifference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.6 Thermoviscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.7 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.8 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

III Hyperbolic Systems of Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1 Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Entropy-Entropy Flux Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 Examples of Hyperbolic Systems of Balance Laws . . . . . . . . . . . 56

3.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

IV The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1 The Cauchy Problem: Classical Solutions . . . . . . . . . . . . . . . . . . . 77

4.2 Breakdown of Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3 The Cauchy Problem: Weak Solutions . . . . . . . . . . . . . . . . . . . . . . 82

4.4 Nonuniqueness of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.5 Entropy Admissibility Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.6 The Vanishing Viscosity Approach . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.7 Initial-Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.8 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

V Entropy and the Stability of Classical Solutions . . . . . . . . . . . . . . . . . 111

5.1 Convex Entropy and the Existence of Classical Solutions . . . . . . 112

5.2 Relative Entropy and the Stability of Classical Solutions . . . . . . 122

5.3 Involutions and Contingent Entropies . . . . . . . . . . . . . . . . . . . . . . . 125

5.4 Contingent Entropies and Polyconvexity . . . . . . . . . . . . . . . . . . . . 138

5.5 The Role of Damping and Relaxation . . . . . . . . . . . . . . . . . . . . . . 146

5.6 Initial-Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

VI The L1 Theory for Scalar Conservation Laws . . . . . . . . . . . . . . . . . . . 175

6.1 The Cauchy Problem: Perseverance and Demise

of Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.2 Admissible Weak Solutions and their Stability Properties . . . . . . 178

6.3 The Method of Vanishing Viscosity . . . . . . . . . . . . . . . . . . . . . . . . 183

6.4 Solutions as Trajectories of a Contraction Semigroup and the

Large Time Behavior of Periodic Solutions . . . . . . . . . . . . . . . . . . 188

6.5 The Layering Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

6.6 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.7 A Kinetic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

6.8 Fine Structure of L¥ Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

6.9 Initial-Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 215

6.10 The L1 Theory for Systems of Conservation Laws . . . . . . . . . . . . 220

6.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

VII Hyperbolic Systems of Balance Laws in One-Space Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

7.1 Balance Laws in One-Space Dimension . . . . . . . . . . . . . . . . . . . . 227

7.2 Hyperbolicity and Strict Hyperbolicity . . . . . . . . . . . . . . . . . . . . . 235

7.3 Riemann Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

7.4 Entropy-Entropy Flux Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

7.5 Genuine Nonlinearity and Linear Degeneracy . . . . . . . . . . . . . . . . 245

7.6 Simple Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

7.7 Explosion of Weak Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

7.8 Existence and Breakdown of Classical Solutions . . . . . . . . . . . . . 253

7.9 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

7.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

VIII Admissible Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

8.1 Strong Shocks, Weak Shocks, and Shocks of Moderate Strength 263

8.2 The Hugoniot Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

<8.3 The Lax Shock Admissibility Criterion;

Compressive, Overcompressive and Undercompressive Shocks . 272

8.4 The Liu Shock Admissibility Criterion . . . . . . . . . . . . . . . . . . . . . 278

8.5 The Entropy Shock Admissibility Criterion . . . . . . . . . . . . . . . . . . 280

8.6 Viscous Shock Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

8.7 Nonconservative Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

8.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

IX Admissible Wave Fans and the Riemann Problem. . . . . . . . . . . . . . . . 303

9.1 Self-Similar Solutions and the Riemann Problem . . . . . . . . . . . . . 303

9.2 Wave Fan Admissibility Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

9.3 Solution of the Riemann Problem via Wave Curves . . . . . . . . . . . 309

9.4 Systems with Genuinely Nonlinear

or Linearly Degenerate Characteristic Families . . . . . . . . . . . . . . . 312

9.5 General Strictly Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . 316

9.6 Failure of Existence or Uniqueness;

Delta Shocks and Transitional Waves . . . . . . . . . . . . . . . . . . . . . . . 320^lt;9.7 The Entropy Rate Admissibility Criterion . . . . . . . . . . . . . . . . . . . 323

9.8 Viscous Wave Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

9.9 Interaction of Wave Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

9.10 Breakdown of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

9.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

X Generalized Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

10.1 BV Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

10.2 Generalized Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

10.3 Extremal Backward Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 362

10.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

XI Scalar Conservation Laws in One Space Dimension . . . . . . . . . . . . . 367

11.1 Admissible BV Solutions and Generalized Characteristics . . . . . 368

11.2 The Spreading of Rarefaction Waves . . . . . . . . . . . . . . . . . . . . . . . 371

11.3 Regularity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

11.4 Divides, Invariants and the Lax Formula . . . . . . . . . . . . . . . . . . . . 377

11.5 Decay of Solutions Induced by Entropy Dissipation . . . . . . . . . . 380

11.6 Spreading of Characteristics and Development of N-Waves . . . . 383

11.7 Confinement of Characteristics

and Formation of Saw-toothed Profiles . . . . . . . . . . . . . . . . . . . . . 384

11.8 Comparison Theorems and L1 Stability . . . . . . . . . . . . . . . . . . . . . 386

11.9 Genuinely Nonlinear Scalar Balance Laws . . . . . . . . . . . . . . . . . . 395

11.10 Balance Laws with Linear Excitation . . . . . . . . . . . . . . . . . . . . . . . 399

11.11 An Inhomogeneous Conservation Law. . . . . . . . . . . . . . . . . . . . . . 401

11.12 When Genuine Nonlinearity Fails . . . . . . . . . . . . . . . . . . . . . . . . . . 406

11.13 Entropy Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

11.14 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

XII Genuinely Nonlinear Systems of Two Conservation Laws . . . . . . . . . 427

12.1 Notation and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

12.2 Entropy-Entropy Flux Pairs and the Hodograph Transformation 429

12.3 Local Structure of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

12.4 Propagation of Riemann Invariants

Along Extremal Backward Characteristics . . . . . . . . . . . . . . . . . . 435

12.5 Bounds on Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

12.6 Spreading of Rarefaction Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

12.7 Regularity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

12.8 Initial Data in L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

12.9 Initial Data with Compact Support . . . . . . . . . . . . . . . . . . . . . . . . . 475

<12.10 Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

12.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

XIII The Random Choice Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

13.1 The Construction Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

13.2 Compactness and Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

13.3 Wave Interactions in Genuinely Nonlinear Systems . . . . . . . . . . . 498

13.4 The Glimm Functional for Genuinely Nonlinear Systems . . . . . . 500

13.5 Bounds on the Total Variation

for Genuinely Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 505

13.6 Bounds on the Supremum for Genuinely Nonlinear Systems . . . 507

13.7 General Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

13.8 Wave Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

13.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

XIV The Front Tracking Method and Standard Riemann Semigroups . . 517

14.1 Front Tracking for Scalar Conservation Laws . . . . . . . . . . . . . . . . 518

14.2 Front Tracking for Genuinely Nonlinear

Systems of Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

14.3 The Global Wave Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

14.4 Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

14.5 Bounds on the Total Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

14.6 Bounds on the Combined Strength of Pseudoshocks . . . . . . . . . . 531

14.7 Compactness and Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

14.8 Continuous Dependence on Initial Data . . . . . . . . . . . . . . . . . . . . . 536

14.9 The Standard Riemann Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . 540

14.10 Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

14.11 Continuous Glimm Functionals,

Spreading of Rarefaction Waves,

and Structure of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

14.12 Stability of Strong Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

14.13 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

XV Construction of BV Solutions by the Vanishing Viscosity Method . . 557

15.1 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557

15.2 Road Map to the Proof of Theorem 15.1.1 . . . . . . . . . . . . . . . . . . . 559

15.3 The Effects of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

15.4 Decomposition into Viscous Traveling Waves . . . . . . . . . . . . . . . . 564

15.5 Transversal Wave Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

15.6 Interaction of Waves of the Same Family . . . . . . . . . . . . . . . . . . . . 572

15.7 Energy Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576

15.8 Stability Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

15.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

XVI BV Solutions for Systems of Balance Laws . . . . . . . . . . . . . . . . . . . . . . 585

16.1 The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

16.2 Strong Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

16.3 Redistribution of Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

16.4 Bounds on the Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

16.5 L1 Stability Via Entropy with Conical Singularity at the Origin . 606

16.6 L1 Stability when the Source is Partially Dissipative . . . . . . . . . . 609

16.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622

XVII Compensated Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

17.1 The Young Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624

17.2 Compensated Compactness and the div-curl Lemma . . . . . . . . . . 625

17.3 Measure-Valued Solutions for Systems of Conservation Laws

and Compensated Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 626

17.4 Scalar Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629

17.5 A Relaxation Scheme for Scalar Conservation Laws . . . . . . . . . . 631

17.6 Genuinely Nonlinear Systems of Two Conservation Laws . . . . . 634

17.7 The System of Isentropic Elasticity . . . . . . . . . . . . . . . . . . . . . . . . 637

17.8 The System of Isentropic Gas Dynamics . . . . . . . . . . . . . . . . . . . . 642<

17.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648

XVIII Steady and Self-similar Solutions in Multi-Space Dimensions . . . . . 655

18.1 Self-Similar Solutions for Multidimensional Scalar

Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655

18.2 Steady Planar Isentropic Gas Flow . . . . . . . . . . . . . . . . . . . . . . . . . 658<

18.3 Self-Similar Planar Irrotational Isentropic Gas Flow . . . . . . . . . . 663

18.4 Supersonic Isentropic Gas Flow Past a Ramp . . . . . . . . . . . . . . . . 667

18.5 Regular Shock Reflection on a Wall . . . . . . . . . . . . . . . . . . . . . . . . 672

18.6 Shock Collision with a Ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675

18.7 Isometric Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678

18.8 Cavitation in Elastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682

18.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813<

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823

O autorze

Professor Dafermos received a Diploma in Civil Engineering from the National Technical University of Athens (1964) and a Ph.D. in Mechanics from the Johns Hopkins University (1967). He has served as Assistant Professor at Cornell University (1968-1971),and as Associate Professor (1971-1975) and Professor (1975- present) in the Division of Applied Mathematics at Brown University. In addition, Professor Dafermos has served as Director of the Lefschetz Center of Dynamical Systems (1988-1993, 2006-2007), as Chairman of the Society for Natural Philosophy (1977-1978) and as Secretary of the International Society for the Interaction of Mathematics and Mechanics. Since 1984, he has been the Alumni-Alumnae University Professor at Brown.

In addition to several honorary degrees, he has received the SIAM W.T. and Idalia Reid Prize (2000), the Cataldo e Angiola Agostinelli Prize of the Accademia Nazionale dei Lincei (2011), the Galileo Medal of the City of Padua (2012), and the Prize of the International Society for the Interaction of Mechanics and Mathematics (2014). He was elected a Fellow of SIAM (2009) and a Fellow of the AMS (2013). In 2016 he received the Wiener Prize, awarded jointly by the American Mathematical Society (AMS) and the Society for Industrial and Applied Mathematics (SIAM).