**mikhailov - partial - differential - equations**

mikhailov - partial - differential - equations

(Parte **1** de 5)

UNITEXT – La Matematica per il 3+2 Volume 9

Editor-in-chief A. Quarteroni

Series editors

L. Ambrosio P. Biscari C. Ciliberto M. Ledoux W.J. Runggaldier

More information about this series at http://www.springer.com/series/5418 More information about this series at http://www.springer.com/series/5418

Sandro Salsa

Partial Differential Equations in Action

From Modelling to Theory

Third Edition

Sandro Salsa Dipartimento di Matematica Politecnico di Milano Milano, Italy

Library of Congress Control Number: 2016932390

© Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproductionon microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Cover illustration: Simona Colombo, Giochi di Graﬁca, Milano, Italy Typesetting with LATEX: PTP-Berlin, Protago TEX-Production GmbH, Germany (w.ptp-berlin.de)

This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

To Anna, my wife To Anna, my wife

Preface

This book is designed as an advanced undergraduate or a ﬁrst-year graduate course for students from various disciplines like applied mathematics, physics, engineering. It has evolved while teaching courses on partial diﬀerential equations (PDEs) during the last few years at the Politecnico di Milano.

The main purpose of these courses was twofold: on the one hand, to train the students to appreciate the interplay between theory and modelling in problems arising in the applied sciences, and on the other hand to give them a solid theoretical background for numerical methods, such as ﬁnite elements.

Accordingly, this textbook is divided into two parts, that we brieﬂy describe below, writing in italics the relevant diﬀerences with the ﬁrst edition, the second one being pretty similar.

The ﬁrst part, Chaps. 2 to 5, has a rather elementary character with the goal of developing and studying basic problems from the macro-areas of diﬀusion, propagation and transport, waves and vibrations. I have tried to emphasize, whenever possible, ideas and connections with concrete aspects, in order to provide intuition and feeling for the subject.

For this part, a knowledge of advanced calculus and ordinary diﬀerential equations is required. Also, the repeated use of the method of separation of variables assumes some basic results from the theory of Fourier series, which are summarized in Appendix A.

Chapter 2 starts with the heat equation and some of its variants in which transport and reaction terms are incorporated. In addition to the classical topics, I emphasized the connections with simple stochastic processes, such as random walks and Brownian motion. This requires the knowledge of some elementary probability. It is my belief that it is worthwhile presenting this topic as early as possible, even at the price of giving up to a little bit of rigor in the presentation. An application to ﬁnancial mathematics shows the interaction between probabilistic and deterministic modelling. The last two sections are devoted to two simple non linear models from ﬂow in porous medium and population dynamics.

viii Preface

Chapter 3 mainly treats the Laplace/Poisson equation. The main properties of harmonic functions are presented once more emphasizing the probabilistic motivations. I have included Perron’s method of sub/super solution, due to is renewed importance as a solution technique for fully non linear equations. The second part of this chapter deals with representation formulas in terms of potentials. In particular, the basic properties of the single and double layer potentials are presented.

Chapter 4 is devoted to ﬁrst order equations and in particular to ﬁrst order scalar conservation laws. The methods of characteristics and the notions of shock and rarefaction waves are introduced through a simple model from traﬃc dynamics. An application to sedimentation theory illustrates the method for non convex/concave ﬂux function. In the last part, the method of characteristics is extended to quasilinear and fully nonlinear equations in two variables.

In Chap. 5 the fundamental aspects of waves propagation are examined, leading to the classical formulas of d’Alembert, Kirchhoﬀ and Poisson. As implem odel of Acoustic Thermography serves as an application of Huygens principle. In the ﬁnal section, the classical model for surface waves in deep water illustrates the phenomenon of dispersion, with the help of the method of stationary phase.

The second part includes the two new Chaps. 9 and 1. In Chaps. 6t o1 0w e develope Hilbert spaces methods for the variational formulation and the analysis of mainly linear boundary and initial-boundary value problems. Given the abstract nature of these chapters, I have made an eﬀort to provide intuition and motivation about the various concepts and results, sometimes running the risk of appearing a bit wordy. The understanding of these topics requires some basic knowledge of Lebesgue measure and integration, summarized in Appendix B.

Chapter 6 contains the tools from functional analysis in Hilbert spaces, necessary for a correct variational formulation of the most common boundary value problems. The main theme is the solvability of abstract variational problems, leading to the Lax-Milgram theorem and Fredholm’s alternative. Emphasis is given to the issues of compactness and weak convergence. Section 6.10 is devoted to the ﬁxed point theorems of Banach and of Schauder and Leray-Schauder.

Chapter 7 is divided into two parts. The ﬁrst one is a brief introduction to the theory of distributions (or generalized functions) of L. Schwartz. In the second one, the most used Sobolev spaces and their basic properties are discussed.

Chapter 8 is devoted to the variational formulation of linear elliptic boundary value problems and their solvability. The development starts with Poisson’s equation and ends with general second order equations in divergence form.

In Chap. 9 I have gathered a number of applications of the variational theory of elliptic equations, in particular to elastostatics and to the stationary Navier-Stokes equations. Also, an application to a simple control problem is discussed.

The issue in Chap. 10, which has been almost completely remodeled,i s the variational formulation of evolution problems, in particular of initial-boundary value problems for second order parabolic operators in divergence form and for the wave equation.

Preface ix

Chapter 1 contains a brief introduction to the basic concepts of the theory of systems of ﬁrst order conservation laws, in one spatial dimension. In particular we extend from the scalar case of Chap 4, the notions of characteristics, shocks, rarefaction waves, contact discontinuity and entropy condition. The main focus is the solution of the Riemann problem.

At the end of each chapter, a number of exercises is included. Some of them can be solved by a routine application of the theory or of the methods developed in the text. Other problems are intended as a completion of some arguments or proofs in the text. Also, there are problems in which the student is required to be more autonomous. The most demanding problems are supplied with answers or hints.

Other (completely solved) exercises can be found in [17], the natural companion of this book by S. Salsa, G. Verzini, Springer 2015.

The order of presentation of the material is clearly a consequence of my |

prejudices. However, the exposition if ﬂexible enough to allow substantial changes without compromising the comprehension and to facilitate a selection of topics for a one or two semester course.

In the ﬁrst part, the chapters are, in practice, mutually independent, with the exception of Subsection 3.3.1 and Sect. 3.4, which presume the knowledge of Sect. 2.6.

In the second part, more attention has to be paid to the order of the arguments.

The material in Sects. 6.1–6.9 and in Sect. 7.1–7.4 and 7.7–7.10 is necessary for understanding the topics in Chap. 8–10. Moreover, Chap. 9 requires also Sect. 6.10, while to cover Chap. 10, also concepts and results from Sect. 7.1 are needed. Finally, Chap. 1 uses Subsections 4.4.2, 4.4.3 and 4.6.1.

Acknowledgments. While writing this book, during the ﬁrst edition, I beneﬁtted from comments and suggestions of many collegues and students.

Among my collegues, I express my gratitude to Luca Dedé, Fausto Ferrari, Carlo

Pagani, Kevin Payne, Alﬁo Quarteroni, Fausto Saleri, Carlo Sgarra, Alessandro Veneziani, Gianmaria A. Verzini and, in particular to Cristina Cerutti, Leonede De Michele and Peter Laurence.

Among the students who have sat through my course on PDEs, I would like to thank Luca Bertagna, Michele Coti-Zelati, Alessandro Conca, Alessio Fumagalli, Loredana Gaudio, Matteo Lesinigo, Andrea Manzoni and Lorenzo Tamellini.

Fo the last two editions, I am particularly indebted to Leonede de Michele,

Ugo Gianazza and Gianmaria Verzini for their interest, criticism and contribution. Many thanks go to Michele Di Cristo, Giovanni Molica-Bisci, Nicola Parolini Attilio Rao and Francesco Tulone for their comments and the time we spent in precious (for me) discussions. Finally, I like to express my appreciation to Francesca Bonadei and Francesca Ferrari of Springer Italia, for their constant collaboration and support.

Milan, April 2016 Sandro Salsa

1 Introduction | 1 |

1.1 Mathematical Modelling | 1 |

1.2 Partial Diﬀerential Equations | 2 |

1.3 Well Posed Problems | 5 |

1.4 Basic Notations and Facts | 7 |

1.5 Smooth and Lipschitz Domains | 12 |

1.6 Integration by Parts Formulas | 15 |

2 Diﬀusion | 17 |

2.1 The Diﬀusion Equation | 17 |

2.1.1 Introduction | 17 |

2.1.2 The conduction of heat | 18 |

2.1.3 Well posed problems (n =1 ) | 20 |

2.1.4 A solution by separation of variables | 23 |

2.1.5 Problems in dimension n> 1 | 32 |

2.2 Uniqueness and Maximum Principles | 34 |

2.2.1 Integral method | 34 |

2.2.2 Maximum principles | 36 |

2.3 The Fundamental Solution | 39 |

2.3.1 Invariant transformations | 39 |

2.3.2 The fundamental solution (n =1 ) | 41 |

2.3.3 The Dirac distribution | 43 |

2.3.4 The fundamental solution (n> 1) | 47 |

2.4 Symmetric Random Walk (n =1 ) | 48 |

2.4.1 Preliminary computations | 49 |

2.4.2 The limit transition probability | 52 |

2.4.3 From random walk to Brownian motion | 54 |

2.5 Diﬀusion, Drift and Reaction | 58 |

2.5.1 Random walk with drift | 58 |

2.5.2 Pollution in a channel | 60 |

Contents 2.5.3 Random walk with drift and reaction . .... .... .... .... ... 63

2.5.4 Critical dimension in a simple population dynamics | 64 |

2.6 Multidimensional Random Walk | 6 |

2.6.1 The symmetric case | 6 |

2.6.2 Walks with drift and reaction | 70 |

2.7 An Example of Reaction–Diﬀusion in Dimension n =3 | 71 |

2.8 The Global Cauchy Problem (n =1 ) | 76 |

2.8.1 The homogeneous case | 76 |

2.8.2 Existence of a solution | 78 |

2.8.3 The nonhomogeneous case. Duhamel’s method | 79 |

2.8.4 Global maximum principles and uniqueness | 82 |

2.8.5 The proof of the existence theorem 2.12 | 85 |

2.9 An Application to Finance | 8 |

2.9.1 European options | 8 |

2.9.2 An evolution model for the price S | 89 |

2.9.3 The Black-Scholes equation | 91 |

2.9.4 The solutions | 95 |

2.9.5 Hedging and self-ﬁnancing strategy | 100 |

2.10 Some Nonlinear Aspects | 102 |

2.10.1 Nonlinear diﬀusion. The porous medium equation | 102 |

2.10.2 Nonlinear reaction. Fischer’s equation | 105 |

Problems | 109 |

3 The Laplace Equation | 115 |

3.1 Introduction | 115 |

3.2 Well Posed Problems. Uniqueness | 116 |

3.3 Harmonic Functions | 118 |

3.3.1 Discrete harmonic functions | 118 |

3.3.2 Mean value properties | 122 |

3.3.3 Maximum principles | 124 |

3.3.4 The Hopf principle | 126 |

3.3.5 The Dirichlet problem in a disc. Poisson’s formula | 127 |

3.3.6 Harnack’s inequality and Liouville’s theorem | 131 |

3.3.7 Analyticity of harmonic functions | 133 |

3.4 A probabilistic solution of the Dirichlet problem | 135 |

3.5 Sub/Superharmonic Functions. The Perron Method | 140 |

3.5.1 Sub/superharmonic functions | 140 |

3.5.2 The method | 142 |

3.5.3 Boundary behavior | 143 |

3.6 Fundamental Solution and Newtonian Potential | 147 |

3.6.1 The fundamental solution | 147 |

3.6.2 The Newtonian potential | 148 |

formula | 151 |

3.7 The Green Function | 155 |

xii Contents 3.6.3 A divergence-curl system. Helmholtz decomposition 3.7.1 An integral identity . .... .... .... .... .... .... .... .... ... 155

3.7.2 Green’s function | 157 |

3.7.3 Green’s representation formula | 160 |

3.7.4 The Neumann function | 161 |

3.8 Uniqueness in Unbounded Domains | 163 |

3.8.1 Exterior problems | 163 |

3.9 Surface Potentials | 166 |

3.9.1 The double and single layer potentials | 166 |

3.9.2 The integral equations of potential theory | 171 |

Problems | 174 |

4 Scalar Conservation Laws and First Order Equations | 179 |

4.1 Introduction | 179 |

4.2 Linear Transport Equation | 180 |

4.2.1 Pollution in a channel | 180 |

4.2.2 Distributed source | 182 |

4.2.3 Extinction and localized source | 183 |

4.2.4 Inﬂow and outﬂow characteristics. A stability estimate | 185 |

4.3 Traﬃc Dynamics | 187 |

4.3.1 A macroscopic model | 187 |

4.3.2 The method of characteristics | 189 |

4.3.3 The green light problem | 191 |

4.3.4 Traﬃc jam ahead | 196 |

4.4 Weak (or Integral) Solutions | 199 |

4.4.1 The method of characteristics revisited | 199 |

4.4.2 Deﬁnition of weak solution | 202 |

condition | 205 |

4.5 An Entropy Condition | 209 |

4.6 The Riemann problem | 212 |

4.6.1 Convex/concave ﬂux function | 212 |

4.6.2 Vanishing viscosity method | 214 |

4.6.3 The viscous Burgers equation | 218 |

4.6.4 Flux function with inﬂection points | 220 |

4.7 An Application to a Sedimentation Problem | 224 |

4.8 The Method of Characteristics for Quasilinear Equations | 230 |

4.8.1 Characteristics | 230 |

4.8.2 The Cauchy problem | 232 |

4.8.3 Lagrange method of ﬁrst integrals | 239 |

4.8.4 Underground ﬂow | 241 |

4.9 General First Order Equations | 244 |

4.9.1 Characteristic strips | 244 |

4.9.2 The Cauchy Problem | 246 |

Contents xiii 4.4.3 Piecewise smooth functions and the Rankine-Hugoniot Problems .... .... .... .... .... .... .... .... .... .... .... .... .... ... 251

5W aves and Vibrations | 259 |

5.1 General Concepts | 259 |

5.1.1 Types of waves | 259 |

5.1.2 Group velocity and dispersion relation | 261 |

5.2 Transversal Waves in a String | 264 |

5.2.1 The model | 264 |

5.2.2 Energy | 266 |

5.3 The One-dimensional Wave Equation | 267 |

5.3.1 Initial and boundary conditions | 267 |

5.3.2 Separation of variables | 269 |

5.4 The d’Alembert Formula | 275 |

5.4.1 The homogeneous equation | 275 |

5.4.2 Generalized solutions and propagation of singularities | 279 |

5.4.3 The fundamental solution | 282 |

5.4.4 Nonhomogeneous equation. Duhamel’s method | 285 |

5.4.5 Dissipation and dispersion | 286 |

5.5 Second Order Linear Equations | 288 |

5.5.1 Classiﬁcation | 288 |

5.5.2 Characteristics and canonical form | 291 |

5.6 The Multi-dimensional Wave Equation (n> 1) | 296 |

5.6.1 Special solutions | 296 |

5.6.2 Well posed problems. Uniqueness | 298 |

5.7 Two Classical Models | 302 |

5.7.1 Small vibrations of an elastic membrane | 302 |

5.7.2 Small amplitude sound waves | 306 |

5.8 The Global Cauchy Problem | 310 |

principle | 310 |

5.8.2 The Kirchhoﬀ formula | 313 |

5.8.3 The Cauchy problem in dimension 2 | 316 |

5.9 The Cauchy Problem with Distributed Sources | 318 |

5.9.1 Retarded potentials (n =3 ) | 318 |

5.9.2 Radiation from a moving point source | 320 |

5.10 An Application to Thermoacoustic Tomography | 324 |

5.1 Linear Water Waves | 328 |

5.1.1 A model for surface waves | 328 |

5.1.2 Dimensionless formulation and linearization | 332 |

5.1.3 Deep water waves | 334 |

5.1.4 Interpretation of the solution | 336 |

5.1.5 Asymptotic behavior | 338 |

5.1.6 The method of stationary phase | 340 |

6 Elements of Functional Analysis | 347 |

6.1 Motivations | 347 |

6.2 Norms and Banach Spaces | 353 |

6.3 Hilbert Spaces | 358 |

6.4 Projections and Bases | 363 |

6.4.1 Projections | 363 |

6.4.2 Bases | 367 |

6.5 Linear Operators and Duality | 373 |

6.5.1 Linear operators | 373 |

6.5.2 Functionals and dual space | 377 |

6.5.3 The adjoint of a bounded operator | 379 |

6.6 Abstract Variational Problems | 382 |

6.6.1 Bilinear forms and the Lax-Milgram Theorem | 382 |

6.6.2 Minimization of quadratic functionals | 387 |

6.6.3 Approximation and Galerkin method | 388 |

6.7 Compactness and Weak Convergence | 391 |

6.7.1 Compactness | 391 |

6.7.2 Compactness in C(Ω) and in Lp(Ω) | 392 |

6.7.3 Weak convergence and compactness | 393 |

6.7.4 Compact operators | 397 |

6.8 The Fredholm Alternative | 399 |

6.8.1 Hilbert triplets | 399 |

6.8.2 Solvability for abstract variational problems | 402 |

6.8.3 Fredholm’s alternative | 405 |

6.9 Spectral Theory for Symmetric Bilinear Forms | 407 |

6.9.1 Spectrum of a matrix | 407 |

6.9.2 Separation of variables revisited | 407 |

6.9.3 Spectrum of a compact self-adjoint operator | 408 |

6.9.4 Application to abstract variational problems | 411 |

6.10 Fixed Points Theorems | 416 |

6.10.1 The Contraction Mapping Theorem | 417 |

6.10.2 The Schauder Theorem | 418 |

6.10.3 The Leray-Schauder Theorem | 420 |

Problems | 421 |

7 Distributions and Sobolev Spaces | 427 |

7.1 Distributions. Preliminary Ideas | 427 |

7.2 Test Functions and Molliﬁers | 429 |

7.3 Distributions | 433 |

7.4 Calculus | 438 |

7.4.1 The derivative in the sense of distributions | 438 |

7.4.2 Gradient, divergence, Laplacian | 440 |

7.5 Operations with Ditributions | 443 |

7.5.1 Multiplication. Leibniz rule | 443 |

7.5.3 Division | 448 |

7.5.4 Convolution | 449 |

7.5.5 Tensor or direct product | 451 |

7.6 Tempered Distributions and Fourier Transform | 454 |

7.6.1 Tempered distributions | 454 |

7.6.2 Fourier transform in S′ | 457 |

7.6.3 Fourier transform in L2 | 460 |

7.7 Sobolev Spaces | 461 |

7.7.1 An abstract construction | 461 |

7.7.2 The space H1 (Ω) | 462 |

7.7.3 The space H10 (Ω) | 466 |

7.7.4 The dual of H10(Ω) | 467 |

7.7.5 The spaces Hm (Ω), m> 1 | 470 |

7.7.6 Calculus rules | 471 |

7.7.7 Fourier transform and Sobolev spaces | 473 |

7.8 Approximations by Smooth Functions and Extensions | 474 |

7.8.1 Local approximations | 474 |

7.8.2 Extensions and global approximations | 475 |

7.9 Traces | 479 |

7.9.1 Traces of functions in H1(Ω) | 479 |

7.9.2 Traces of functions in Hm(Ω) | 483 |

7.9.3 Trace spaces | 484 |

7.10 Compactness and Embeddings | 487 |

7.10.1 Rellich’s theorem | 487 |

7.10.2 Poincaré’s inequalities | 488 |

7.10.3 Sobolev inequality in Rn | 490 |

7.10.4 Bounded domains | 492 |

7.1 Spaces Involving Time | 494 |

7.1.1 Functions with values into Hilbert spaces | 494 |

7.1.2 Sobolev spaces involving time | 497 |

Problems | 499 |

8 Variational Formulation of Elliptic Problems | 505 |

8.1 Elliptic Equations | 505 |

8.2 Notions of Solutions | 507 |

8.3 Problems for the Poisson Equation | 509 |

8.3.1 Dirichlet problem | 509 |

8.3.2 Neumann, Robin and mixed problems | 512 |

8.3.3 Eigenvalues and eigenfunctions of the Laplace operator | 517 |

8.3.4 An asymptotic stability result | 519 |

8.4 General Equations in Divergence Form | 521 |

8.4.1 Basic assumptions | 521 |

8.4.2 Dirichlet problem | 522 |

8.4.3 Neumann problem | 527 |

8.5 Weak Maximum Principles | 531 |

8.6 Regularity | 536 |

Problems | 544 |

9 Further Applications | 551 |

9.1 A Monotone Iteration Scheme for Semilinear Equations | 551 |

9.2 Equilibrium of a Plate | 554 |

9.3 The Linear Elastostatic System | 556 |

9.4 The Stokes System | 561 |

9.5 The Stationary Navier Stokes Equations | 566 |

9.5.1 Weak formulation and existence of a solution | 566 |

9.5.2 Uniqueness | 569 |

9.6 A Control Problem | 571 |

9.6.1 Structure of the problem | 571 |

9.6.2 Existence and uniqueness of an optimal pair | 572 |

9.6.3 Lagrange multipliers and optimality conditions | 574 |

9.6.4 An iterative algorithm | 575 |

Problems | 576 |

10 Weak Formulation of Evolution Problems | 581 |

10.1 Parabolic Equations | 581 |

10.2 The Cauchy-Dirichlet Problem for the Heat Equation | 583 |

10.3 Abstract Parabolic Problems | 586 |

10.3.1 Formulation | 586 |

10.3.2 Energy estimates. Uniqueness and stability | 589 |

10.3.3 The Faedo-Galerkin approximations | 591 |

10.3.4 Existence | 592 |

10.4 Parabolic PDEs | 593 |

10.4.1 Problems for the heat equation | 593 |

10.4.2 General Equations | 596 |

10.4.3 Regularity | 598 |

10.5 Weak Maximum Principles | 600 |

10.6 The Wave Equation | 602 |

10.6.1 Hyperbolic Equations | 602 |

10.6.2 The Cauchy-Dirichlet problem | 603 |

10.6.3 The method of Faedo-Galerkin | 605 |

10.6.4 Solution of the approximate problem | 606 |

10.6.5 Energy estimates | 607 |

10.6.6 Existence, uniqueness and stability | 609 |

Problems | 611 |

1 Systems of Conservation Laws | 615 |

1.1 Introduction | 615 |

1.2 Linear Hyperbolic Systems | 620 |

1.2.2 Classical solutions of the Cauchy problem | 621 |

Riemann problem | 623 |

1.3 Quasilinear Conservation Laws | 627 |

1.3.1 Characteristics and Riemann invariants | 627 |

condition | 630 |

1.4 The Riemann Problem | 631 |

wave | 636 |

discontinuities | 638 |

1.4.4 Solution of the Riemann problem by a single k-shock | 640 |

1.4.5 The linearly degenerate case | 642 |

1.4.6 Local solution of the Riemann problem | 643 |

1.5 The Riemann Problem for the p-system | 644 |

1.5.1 Shock waves | 644 |

1.5.2 Rarefaction waves | 646 |

1.5.3 The solution in the general case | 649 |

Problems | 653 |

Appendix A. Fourier Series | 657 |

A.1 Fourier Coeﬃcients | 657 |

A.2 Expansion in Fourier Series | 660 |

Appendix B. Measures and Integrals | 663 |

B.1 Lebesgue Measure and Integral | 663 |

B.1.1 A counting problem | 663 |

B.1.2 Measures and measurable functions | 665 |

B.1.3 The Lebesgue integral | 667 |

B.1.4 Some fundamental theorems | 668 |

integrals | 670 |

Appendix C. Identities and Formulas | 673 |

C.1 Gradient, Divergence, Curl, Laplacian | 673 |

C.2 Formulas | 675 |

References | 677 |

xviii Contents 1.2.3 Homogeneous systems with constant coeﬃcients. The 1.3.2 Weak (or integral) solutions and the Rankine-Hugoniot 1.4.1 Rarefaction curves and waves. Genuinely nonlinear systems .633 1.4.2 Solution of the Riemann problem by a single rarefaction 1.4.3 Lax entropy condition. Shock waves and contact B.1.5 Probability spaces, random variables and their Index . .. .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... 681

Chapter 1 Introduction

1.1 Mathematical Modelling

Mathematical modelling plays a big role in the description of a large part of phenomena in the applied sciences and in several aspects of technical and industrial activity.

By a “mathematical model” we mean a set of equations and/or other mathematical relations capable of capturing the essential features of a complex natural or artiﬁcial system, in order to describe, forecast and control its evolution. The applied sciences are not conﬁned to the classical ones; in addition to physics and chemistry, the practice of mathematical modelling heavily aﬀects disciplines like ﬁnance, biology, ecology, medicine, sociology.

In the industrial activity (e.g. for aerospace or naval projects, nuclear reactors, combustion problems, production and distribution of electricity, traﬃc control, etc.) the mathematical modelling, involving ﬁrst the analysis and the numerical simulation and followed by experimental tests, has become a common procedure, necessary for innovation, and also motivated by economic factors. It is clear that all of this is made possible by the enormous computational power now available.

In general, the construction of a mathematical model is based on two main ingredients:

general laws and constitutive relations.

In this book we shall deal with general laws coming from continuum mechanics and appearing as conservation or balance laws (e.g. of mass, energy, linear momentum, etc.).

The constitutive relations are of an experimental nature and strongly depend on the features of the phenomena under examination. Examples are the Fourier law of heat conduction, the Fick’s law for the diﬀusion of a substance or the way the speed of a driver depends on the density of cars ahead.

The outcome of the combination of the two ingredients is usually a partial diﬀerential equation or a system of them.

© Springer International Publishing Switzerland 2016 S. Salsa, Partial Diﬀerential Equations in Action. From Modelling to Theory,3 rd Ed., UNITEXT – La Matematica per il 3+2 9, DOI 10.1007/978-3-319-31238-5_1

2 1 Introduction 1.2 Partial Diﬀerential Equations

F (x1, | ,x n,u ,u x ,... ,u x ,u x x ,u x x ... ,u x x ,u x x x ,... )=0 (1.1) |

where the unknown u = u(x1, | xn) is a function of n variables and ux ,... , |

ux x , | are its partial derivatives. The highest order of diﬀerentiation occurring |

A partial diﬀerential equation is a relation of the following type: in the equation is the order of the equation.

A ﬁrst important distinction is between linear and nonlinear equations. Equation (1.1) is linear if F is linear with respect to u and all its derivatives, otherwise it is nonlinear. A second distinction concerns the types of nonlinearity. We distinguish:

• Semilinear equations when F is nonlinear only with respect to u but is linear with respect to all its derivatives, with coeﬃcients depending only on x.

• Quasi-linear equations when F is linear with respect to the highest order derivatives of u, with coeﬃcients depending only on x, u and lower order derivatives.

• Fully nonlinear equations when F is nonlinear with respect to the highest order derivatives of u.

The theory of linear equations can be considered suﬃciently well developed and consolidated, at least for what concerns the most important questions. On the contrary, the nonlinearities present such a rich variety of aspects and complications that a general theory does not appear to be conceivable. The existing results and the new investigations focus on more or less speciﬁc cases, especially interesting in the applied sciences.

To give the reader an idea of the wide range of applications we present a series of examples, suggesting one of the possible interpretations. Most of them are considered at various level of deepness in this book. In the examples, x represents a space variable (usually in dimension n =1 ,2,3)a nd t is a time variable.

We start with linear equations. In particular, equations (1.2)–(1.5) are fundamental and their theory constitutes a starting point for many other equations.

1. Transport equation (ﬁrst order):

It describes for instance the transport of a solid polluting substance along a channel; here u is the concentration of the substance and v is the stream speed. We consider the one-dimensional version of (1.2) in Sect. 4.2.

(Parte **1** de 5)