From Particle Systems to Partial Differential Equations: PSPDE V, Braga, Portugal, November 2016 (Springer Proceedings in Mathematics & Statistics Book 258)

This book presents the proceedings of the international conference Particle Systems and Partial Differential Equations V, which was held at the University of Minho, Braga, Portugal, from the 28th to 30th November 2016. It includes papers on mathematical problems motivated by various applications in physics, engineering, economics, chemistry, and biology. The purpose of the conference was to bring together prominent researchers working in the fields of particle systems and partial differential equations, providing a venue for them to present their latest findings and discuss their areas of expertise. Further, it was intended to introduce a vast and varied public, including young researchers, to the subject of interacting particle systems, its underlying motivation, and its relation to partial differential equations. The book appeals to probabilists, analysts and also to mathematicians in general whose work focuses on topics in mathematical physics, stochastic processes and differential equations, as well as to physicists working in the area of statistical mechanics and kinetic theory.-- Provided by publisher Preface 6 Contents 7 Linear Boltzmann Equations: A Gradient Flow Formulation 8 1 Introduction 8 2 A Gradient Flow Formulation of Heat Equation and Linear Boltzmann Equation 10 2.1 Heat Equation 10 2.2 Linear Boltzmann Equations 13 3 Diffusive Scaling 16 References 18 Navier–Stokes Hydrodynamic Limit of BGK Kinetic Equations for an Inert Mixture of Polyatomic Gases 20 1 Introduction 20 2 BGK Model 22 3 Hydrodynamic Limit: Conserved Quantities and Leading Order Accuracy 24 4 First Order Distributions fs(1) 26 5 Asymptotic Closure and Navier–Stokes Equations 29 5.1 Computation of Number Densities 29 5.2 Computation of Diffusion Velocities 30 5.3 Computation of Dynamical Pressure and of Viscous Stress Tensor 32 5.4 Computation of Thermal Heat Flux 35 6 Conclusion and Perspectives 36 References 37 Quantization of Probability Densities: A Gradient Flow Approach 39 1 Introduction 39 2 A Compendium of Quantization Theory 40 3 The Gradient Flow Approach 43 4 The One-Dimensional Case 45 4.1 Computing FN,r in the One-Dimensional Setting 45 4.2 The Slowly Varying Setting 46 4.3 The Continuous Functional mathcalFr and its L2-Gradient Flow 47 4.4 The Eulerian Formulation of the Gradient Flow 48 4.5 Gradient Structure of the Eulerian Formulation (4) 49 4.6 Main Results in the One-Dimensional Case 50 5 The Two-Dimensional Case 52 6 Final Remarks 57 References 58 Semi-Lagrangian Approximation of BGK Models for Inert and Reactive Gas Mixtures 59 1 Introduction 59 2 Kinetic Boltzmann-Type Equations 61 3 BGK Model Preserving Exchange Rates 63 3.1 The BGK Model of Andries, Aoki and Perthame (AAP) 63 3.2 The Extension to a Chemically Reacting Mixture 65 4 BGK Models Preserving Global Conservations 66 4.1 Relaxation Model for Inert Mixtures 67 4.2 Extension to the Reactive Case 70 5 Lagrangian Formulation of the BGK Equation and Numerical Schemes 71 5.1 First Order Scheme 72 5.2 Second Order BDF Method 73 6 Numerical Approximation of BGK Models for Mixtures 74 6.1 First Order Semi-Lagrangian Scheme for the AAP BGK Model 75 6.2 Sketch of the First Order Semi-Lagrangian Scheme for the Reactive BGK Model 77 7 Numerical Results 78 References 85 Hydrostatic Limit and Fick's Law for the Symmetric Exclusion with Long Jumps 87 1 Introduction 87 2 Notation and Results 89 2.1 The Model 89 2.2 Hydrostatic Equation 91 2.3 Statement of Results 94 3 Hydrostatic Limit and Fick's Law 96 3.1 Proof of Theorem 1 99 3.2 Proof of Theorem 2 101 4 Proof of Theorem 3 103 References 110 Hydrodynamic Analysis of Sound Wave Propagation in a Reactive Mixture Confined Between Two Parallel Plates 111 1 Introduction 112 2 Description of the Mixture 113 3 Statement of the Problem 114 4 Hydrodynamic Equations for the Reactive Mixture 116 5 Analysis of Sound Propagation in the Reactive Mixture 118 6 Results and Discussion 122 7 Final Remarks and Future Plans 126 References 127 Porous Medium Model in Contact with Slow Reservoirs 129 1 Introduction 129 2 Statement of Results 132 2.1 The Model 132 2.2 Hydrodynamic Equations 135 2.3 Hydrodynamic Limit 137 3 Discrete Versions of Weak Solutions 139 4 Tightness 143 5 Auxiliary Results 146 5.1 Dirichlet Forms 146 5.2 Replacement Lemmas 149 References 153 On the Fibonacci Universality Classes in Nonlinear Fluctuating Hydrodynamics 154 1 Introduction 154 2 Nonlinear Fluctuating Hydrodynamics 156 2.1 Notation and General Properties of Fluctuations 156 2.2 Nonlinear Fluctuating Hydrodynamics 158 2.3 Mode Coupling Theory 159 2.4 Fibonacci Universality Classes 160 3 Two-Lane Lattice Gas for the Modified KPZ Universality Class 161 4 Criterion for Lévy Universality Classes for Systems with Two Conservation Laws 165 4.1 Diagonalization of the Current Jacobian 166 4.2 Non-KPZ Universality Classes 166 References 170 Front Matter ....Pages i-vii Linear Boltzmann Equations: A Gradient Flow Formulation (Giada Basile)....Pages 1-12 Navier–Stokes Hydrodynamic Limit of BGK Kinetic Equations for an Inert Mixture of Polyatomic Gases (Marzia Bisi, Giampiero Spiga)....Pages 13-31 Quantization of Probability Densities: A Gradient Flow Approach (François Golse)....Pages 33-52 Semi-Lagrangian Approximation of BGK Models for Inert and Reactive Gas Mixtures (M. Groppi, G. Russo, G. Stracquadanio)....Pages 53-80 Hydrostatic Limit and Fick’s Law for the Symmetric Exclusion with Long Jumps (Byron Jiménez Oviedo, Arthur Vavasseur)....Pages 81-104 Hydrodynamic Analysis of Sound Wave Propagation in a Reactive Mixture Confined Between Two Parallel Plates (Denize Kalempa, Adriano W. Silva, Ana Jacinta Soares)....Pages 105-122 Porous Medium Model in Contact with Slow Reservoirs (Renato de Paula, Patrícia Gonçalves, Adriana Neumann)....Pages 123-147 On the Fibonacci Universality Classes in Nonlinear Fluctuating Hydrodynamics (G. M. Schütz)....Pages 149-167