From Particle Systems to Partial Differential Equations: International Conference, Particle Systems and PDEs VI, VII and VIII, 2017-2019 (Springer Proceedings in Mathematics & Statistics, 352)

This book includes the joint proceedings of the International Conference on Particle Systems and PDEs VI, VII and VIII. Particle Systems and PDEs VI was held in Nice, France, in November/December 2017, Particle Systems and PDEs VII was held in Palermo, Italy, in November 2018, and Particle Systems and PDEs VIII was held in Lisbon, Portugal, in December 2019. Most of the papers are dealing with mathematical problems motivated by different applications in physics, engineering, economics, chemistry and biology. They illustrate methods and topics in the study of particle systems and PDEs and their relation. The book is recommended to probabilists, analysts and to those mathematicians in general, whose work focuses on topics in mathematical physics, stochastic processes and differential equations, as well as to those physicists who work in statistical mechanics and kinetic theory. Preface Contents Brief Discussion of the Lr-Theory for the Boltzmann Equation: Cutoff and Non-cutoff 1 Introduction 2 The Cutoff Case 2.1 Propagation of the Lr-Norm for the Cutoff Case 2.2 Propagation of the Linfty-Norm for the Cutoff Case 3 The Non-cutoff Case 3.1 The Lr-Theory for the Non-cutoff Case 3.2 The Linfty-Theory for the Non-cutoff Case References The Maxwell–Stefan Diffusion Limit of a Hard-Sphere Kinetic Model for Mixtures 1 Introduction 2 Kinetic Model 3 Scaled Kinetic Equations, Properties and Assumptions 3.1 Scaled Kinetic Equations 3.2 Properties of the Collision Operators 3.3 Assumptions 4 The Maxwell–Stefan Diffusion Limit 4.1 Continuity Equations for the Species 4.2 Momentum Balance Equations for the Species 4.3 Formal Asymptotics 5 Conclusion References An Asymptotic Preserving Scheme for a Stochastic Linear Kinetic Equation in the Diffusion Regime 1 Introduction 2 General Setting 3 The Diffusion Limit 3.1 The Macroscopic Equation 3.2 The Micro-Macro Decomposition 3.3 Formal Analytical Limit 4 The Numerical Scheme 4.1 Construction of the Scheme 4.2 Formal Numerical Limit 4.3 Stability Results 5 Numerical Tests 6 Conclusion and Perspectives References Zero-Range Process in Random Environment 1 Introduction 2 A Preliminary Illustration: Traffic Jams 3 Description of the Model, Basic Properties 3.1 The Process and Its Invariant Measures 3.2 Assumptions on the Environment, and Consequences 4 Convergence 4.1 Previous Results for Convergence 4.2 Our Results for Convergence 5 Hydrodynamics 5.1 Previous Results for Hydrodynamic Limit 5.2 Our Results on Hydrodynamic Limits 5.3 Examples 6 Local Equilibrium References On Non-equilibrium Fluctuations for the Stirring Process with Births and Deaths 1 Introduction 2 Preliminaries 2.1 The Model 2.2 Hydrodynamic Limit 2.3 Fluctuations 3 Closure of the Martingale 4 Macroscopic Limit of the Fluctuation Field 4.1 Direct Calculation of the Variance 4.2 Continuous Kernel Variance Using Martingales References On Existence, Uniqueness and Banach Space Regularity for Solutions of Boltzmann Equations Systems for Monatomic Gas Mixtures 1 Introduction 2 Kinetic Model 2.1 Collision Process 2.2 The System of Boltzmann Equations 2.3 Representations of the Collision Operator 3 Moments and Functional Space 4 Statement of the Problem 5 Angular Averaging Estimate for the Mixture System Gain Operators 6 Polynomially and Exponentially Weighted L1 Theory 6.1 Polynomially Weighted L1 Norms 6.2 Exponentially Weighted L1 Norms 7 Existence and Uniqueness Theory 8 Polynomial and Exponential Weighted Lp Theory, 1