Nonlinear PDEs, Their Geometry, and Applications: Proceedings of the Wisła 18 Summer School (Tutorials, Schools, and Workshops in the Mathematical Sciences)

This volume presents lectures given at the Summer School Wisła 18: Nonlinear PDEs, Their Geometry, and Applications, which took place from August 20 - 30th, 2018 in Wisła, Poland, and was organized by the Baltic Institute of Mathematics. The lectures in the first part of this volume were delivered by experts in nonlinear differential equations and their applications to physics. Original research articles from members of the school comprise the second part of this volume. Much of the latter half of the volume complements the methods expounded in the first half by illustrating additional applications of geometric theory of differential equations. Various subjects are covered, providing readers a glimpse of current research. Other topics covered include thermodynamics, meteorology, and the Monge–Ampère equations. Researchers interested in the applications of nonlinear differential equations to physics will find this volume particularly useful. A knowledge of differential geometry is recommended for the first portion of the book, as well as a familiarity with basic concepts in physics. Foreword Preface Acknowledgements Contents Contributors Acronyms Part I Lectures 1 Contact Geometry, Measurement, and Thermodynamics 1.1 Preface 1.2 A Crash Course in Probability Theory 1.2.1 Measure Spaces and Measurable Maps 1.2.2 Operations Over Measures, Measure Spaces, and Measurable Maps 1.2.3 The Lebesgue Integral 1.2.4 The Radon–Nikodym Theorem 1.2.5 The Fubini Theorem 1.2.6 Random Vectors 1.2.7 Conditional Expectation 1.2.8 Dependency, Coherence Conditions, and Tensor Product of Random Vectors 1.3 Measurement of Random Vectors 1.3.1 Entropy and the Shannon Formula 1.3.2 Gain of Information 1.3.3 Principle of Minimal Information Gain 1.3.4 The Gaussian Distribution 1.3.5 Central Moments 1.3.6 Change of Information Gain 1.3.7 Constraints and Constitutive Relations 1.3.8 Application to Classical Mechanics and Classical Field Theory 1.4 Thermodynamics 1.4.1 Laws of Thermodynamics 1.4.2 Thermodynamics and Measurement 1.4.3 Gases 1.4.4 Thermodynamic Processes and Contact Transformations References 2 Lectures on Geometry of Monge–Ampère Equations with Maple 2.1 Introduction 2.2 Lecture 1. Introduction to Contact Geometry 2.2.1 Bundle of 1-Jets 2.2.2 Contact Transformations 2.3 Lecture 2. Geometrical Approach to Monge–Ampère Equations 2.3.1 Non-linear Second-Order Differential Operators 2.3.2 Multivalued Solutions of Monge–Ampère Equations 2.3.3 Effective Forms 2.4 Lecture 3. Contact Transformations of Monge–Ampère Equations 2.5 Lecture 4. Geometrical Structures 2.5.1 Pfaffians 2.5.2 Fields of Endomorphisms 2.5.3 Characteristic Distributions 2.5.4 Symplectic Monge–Ampère Equations 2.5.5 Splitting of Tangent Spaces 2.6 Lecture 5. Tensor Invariants of Monge–Ampère Equations 2.6.1 Decomposition of de Rham Complex 2.6.2 Tensor Invariants 2.6.3 The Laplace Forms 2.6.4 Contact Linearization of the Monge–Ampère Equations References 3 Geometry of Monge–Ampère Structures 3.1 About These Lectures 3.2 Lecture One: What Is It All About? 3.2.1 Basic Geometric Structures 3.2.2 Kähler, Special and Other Related Structures 3.2.3 Holomorphic Symplectic Structures 3.2.4 Lagrangian, Special Lagrangian and Complex Lagrangian Submanifolds 3.2.5 Hyperkähler Manifolds 3.2.6 Generalised Complex Structure 3.2.7 Notes and Further Reading 3.3 Lecture Two: Recursion (Nijenuijs) Operators and Some Related Algebraic Constructions 3.3.1 Recursion Operators and Its Properties 3.3.2 Triples of Symplectic Forms 3.3.3 Notes and Further Reading 3.4 Lecture Three: Symplectic Monge–Ampère Operators and Equations 3.4.1 Monge–Ampère Equations 3.4.2 Geometry of Differential Forms 3.4.3 Notes and Further Reading 3.5 Lecture Four: Monge–Ampère Structures 3.5.1 General Properties 3.5.2 (4m+2)-Dimensional MA Geometry 3.5.3 Explicite Examples of Generalised Almost Calabi–Yau on T*mathbbR3 (After B.Banos) 3.5.4 Notes and Further Reading 3.6 Lecture Five 3.6.1 Bi-Lagrangian, Special Lagrangian, Special Kähler and Monge–Ampère Equations 3.6.2 2d and 3d Rotating Stratified Flows—Dritschel–Viudez Diagnostic MAEs 3.6.3 Generalised Complex Geometry and Monge–Ampère Structures 3.6.4 Notes and Further Reading References 4 Introduction to Symbolic Computations in Differential Geometry with Maple 4.1 Introduction 4.2 Basic Setup 4.2.1 Subpackage Tools 4.3 Calculations with Vectors and Forms 4.3.1 Computing Symmetries 4.4 Transformations 4.4.1 Operations on Transformations Reference Part II Participants Contributions 5 On the Geometry Arising in Some Meteorological Models in Two and Three Dimensions 5.1 Introduction 5.2 A Brief Guide in Balanced Meteorological Models 5.3 Monge–Ampère Geometry 5.3.1 Monge–Ampère Operators 5.3.2 Generalized Solutions 5.3.3 The Problem of Local Equivalence 5.3.4 Monge–Ampère Structures 5.4 2d Rotating Stratified Flows—Dritschel–Viudez MAE 5.4.1 Integrability of the Complex/Product Structure 5.4.2 Underlying Hypersymplectic Geometry 5.4.3 2d-Diagnostic Equation of Dritschel–Viudez: Special Choice of Constant Coefficients 5.4.4 Reduction to Constant Coefficients 5.4.5 Variation of the Potential and Hyper-Kähler Metrics in 4d 5.5 Some Examples of 3d-Geostrophic Models 5.5.1 The Birkett and Thorpe Equation (BT) 5.5.2 The Hoskins Equation (H) 5.5.3 The McIntyre-Roulstone Equation (McI) 5.5.4 The Snyder–Skamarock–Rotunno Equation (SSR) 5.6 3d Rotating Stratified Flows—Dritschel–Viudez MAE References 6 Gas Flow with Phase Transitions: Thermodynamics and the Navier–Stokes Equations 6.1 Introduction 6.2 Geometric Representation of Thermodynamic States 6.3 Van der Waals Gases 6.3.1 The Equations of State 6.3.2 Applicable Domains for the Van der Waals Gas 6.3.3 Phase Transitions 6.4 Asymptotic Expansions for Solution 6.4.1 Zeroth-Order Approximation 6.4.2 First-Order Approximation 6.5 Phase Transitions Along the Gas Flow References 7 Differential Invariants in Thermodynamics 7.1 Introduction 7.2 Geometry of Thermodynamics 7.3 Equivalence of Thermodynamical Systems 7.4 Differential Invariants 7.4.1 Differential Invariants Under Aff(V) 7.4.2 Differential Invariants Under G0 timesAff(V) 7.5 Application to Gases 7.5.1 Distinguishing Gases 7.5.2 Ideal Gas 7.5.3 Van der Waals Gas References 8 Monge–Ampère Grassmannians, Characteristic Classes and All That 8.1 Grassmannians, Associated with the Lagrangian and Legendrian Planes 8.2 Integral or Monge–Ampère Grassmannians 8.3 Grassmannians for 2- and 3- Effective Forms 8.3.1 Integral Grassmannians for Monge–Ampère Equations in Dimension 2 8.3.2 Geometric Structure Associated with 3d- MA Equations 8.3.3 Integrability and MA Grassmannians in 3d 8.4 Multidimensional Generalisation of Splitting Construction 8.4.1 Non-degenerate 2k+1- Forms in Sense of Hitchin 8.5 Characteristic Classes of Monge–Ampère Equations on a 3-Dimensional Manifolds 8.5.1 Special Lagrangian Monge–Ampère Characteristic Classes 8.5.2 Remarks and Speculations About mathbbS References 9 Weak Inverse Problem of Calculus of Variations for Geodesic Mappings and Relation to Harmonic Maps 9.1 Geodesic Mappings and Basic Setting 9.2 Harmonic Mappings 9.3 Weak Inverse Problem of Calculus of Variations 9.4 Summary and Conclusions References 10 Integrability of Geodesics of Totally Geodesic Metrics 10.1 Introduction 10.2 Singularities 10.3 Geodesics 10.3.1 Geodesic Equations 10.3.2 Integrability of Geodesic Equations 10.4 Einstein–Maxwell Solutions 10.5 Discussion 10.6 Conclusions References Index This volume presents lectures given at the Summer School Wisła 18: Nonlinear PDEs, Their Geometry, and Applications, which took place from August 20 - 30th, 2018 in Wisła, Poland, and was organized by the Baltic Institute of Mathematics. The lectures in the first part of this volume were delivered by experts in nonlinear differential equations and their applications to physics. Original research articles from members of the school comprise the second part of this volume. Much of the latter half of the volume complements the methods expounded in the first half by illustrating additional applications of geometric theory of differential equations. Various subjects are covered, providing readers a glimpse of current research. Other topics covered include thermodynamics, meteorology, and the Monge-Ampère equations. Researchers interested in the applications of nonlinear differential equations to physics will find this volume particularly useful. A knowledge of differential geometry is recommended for the first portion of the book, as well as a familiarity with basic concepts in physics.-- Provided by publisher