دانلود کتاب 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)

فهرست مطالب :

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 9 Polynomial and Exponential Linfty Theory
References
Hydrodynamics of Weakly Asymmetric Exclusion with Slow Boundary
1 Introduction
2 Statement of Results
2.1 The Models
2.2 Hydrodynamic Equations
2.3 Hydrodynamic Limit
3 Heuristics for Hydrodynamic Equations
4 Tightness
5 Replacement Lemmas
5.1 Estimates on Dirichlet Forms
5.2 Proof of Lemma 5
5.3 Proof of Lemma 6
Appendix
References
Recent Developments on the Modelling of Cell Interactions in Autoimmune Diseases
1 Introduction
2 The Biological Scenario
3 The Model for Autoimmune Diseases
3.1 The Kinetic Description of the Cellular Interactions
3.2 The Macroscopic Equations
3.3 The Wellposedeness of the Model
3.4 Numerical Solutions of the Model
4 The Extended Model with Immunotherapy
4.1 The Model
4.2 Mathematical Analysis and Optimal Control
4.3 Numerical Solutions of the Model with Immunotherapy
5 Conclusion and Future Projects
References
Geometrical Structures of the Instantaneous Current and Their Macroscopic Effects: Vortices and Perspectives in Non-gradient Models
1 Introduction and Results
2 Definitions
3 Particle Models and Instantaneous Current
3.1 Exclusion Process and the 2-SEP
3.2 Instantaneous Current in Particle Systems
4 Energy-Mass Models
4.1 KMP Model and Generalization, Dual KMP, Gaussian Model
4.2 Weakly Asymmetric Energy-Mass Models
4.3 Instantaneous Current of Energy-Mass Systems
5 Discrete Hodge Decomposition in Interacting Particle Systems
5.1 Functional Discrete Hodge Decomposition and Lattice Gases
5.2 The One Dimensional Case
5.3 The Two Dimensional Case
6 Interacting Particle Systems with Vorticity
7 Scaling Limits and Transport Coefficients of Diffusive Models
7.1 Qualitative Derivation of Hydrodynamics
8 Scaling Limit of an Exclusion Process with Vorticity
9 Green-Kubo\'s Formula and Perspectives in Non-gradient Particles Systems
References
Porous Medium Model: An Algebraic Perspective and the Fick\'s Law
1 Introduction
1.1 Organization of the Paper
1.2 The Model
2 Fick\'s Law for the PMM with Slow Reservoirs
2.1 Currents
2.2 Integrated Currents
2.3 Empirical Measures
2.4 Fick\'s Law
2.5 Proof of Theorem 1
3 Stochastic Duality Relations for the PMM
3.1 Algebraic Approach to Duality
3.2 Porous Medium Model Described with the mathfraksu(2) Algebra
3.3 Duality Relations for the Porous Medium Model
References
Forward Utilities and Mean-Field Games Under Relative Performance Concerns
1 Introduction
2 Asset Specialization, Forward Utilities and CARA Preferences
2.1 Forward Dynamic Utilities (Classic)
3 Forward Relative Performance Criteria
3.1 Forward Relative Performance Criteria
3.2 The Forward Nash Equilibrium
4 The Mean Field Game
4.1 Agents Through Type-Distribution and the Market
4.2 The Equilibrium
4.3 Solving the Optimization Problem
4.4 Mean-Field Dynamic Model Selection with Large Horizons
5 Outlook and Open Questions
6 Supplementary Calculations
References
The Boundary Driven Zero-Range Process
1 Introduction
2 Definition of the Model
3 Invariant Measure
4 Hydrostatic Limit
5 Attractiveness
6 Tightness
6.1 Related Martingales
6.2 Proof of Tightness
7 Limit Points are Concentrated on Absolutely Continuous Measures
8 Hydrodynamic Limit
9 Heuristics of the Hydrodynamic Equation
10 Heuristics for Hydrodynamics of the General Model
References
Partial Regularity in Time for the Landau Equation (with Coulomb Interaction)
1 The Landau Equation
2 A Notion of Weak Solutions of the Landau Equation
2.1 Villani\'s H-Solutions
2.2 Suitable Solutions
3 Partial Regularity in t for Suitable Solutions
4 Existence Theory for Suitable Solutions
4.1 Formal H Theorem
4.2 The Desvillettes Theorem
4.3 Sketch of the Proof of Proposition 1
5 The 1st De Giorgi Type Lemma
6 The Improved De Giorgi Lemma
7 Proof of the Partial Regularity Theorem
8 Final Remarks and Open Problems
References
Recent Developments on the Well-Posedness Theory for Vlasov-Type Equations
1 An Introduction to Vlasov-Type Equations in Plasma Physics
1.1 The Vlasov–Poisson System: The Electrons\' View-Point
1.2 The Vlasov–Poisson System with Massless Electrons: The Ions\' View-Point
2 Well-Posedness for Vlasov Equations with Smooth Interactions
3 Well-Posedness for the Vlasov–Poisson System
4 Well-Posedness Theory for the Vlasov–Poisson System with Massless Electrons
4.1 Strategy for mathbbTd
4.2 Strategy for mathbbR3
References
Charge-Current Correlation Identities for Stochastic Interacting Particle Systems
1 Introduction
2 Classical Setting and Results
2.1 Notation and General Properties of the Interacting Particle System
2.2 Some Important Expectation Values and General Properties of the Invariant Measure
2.3 Results
3 Proofs
3.1 Proposition 1
3.2 Theorems 1 and 2
3.3 Theorem 3
4 Comments on Phase Separation and the Decay of Correlations
References
From the Hartree to the Vlasov Dynamics: Conditional Strong Convergence
1 Introduction
2 Preliminary Estimates
3 Proof of Theorem 1
References
From the Boltzmann Description for Mixtures to the Maxwell–Stefan Diffusion Equations
1 Introduction
2 The Maxwell–Stefan Equations
3 The Boltzmann System for Monatomic Inert Gaseous Mixtures
4 The Boltzmann System for Polyatomic Reactive or Non-reactive Gas Mixtures
5 The Maxwell–Stefan Diffusion Limit for Non-reactive Boltzmann Systems
5.1 The Formal Asymptotics
5.2 The Rigorous Diffusive Asymptotics
6 The Maxwell–Stefan Diffusion Limit for Reactive Kinetic Systems
7 Some Related Research Directions
References
Alternative Quantum Formulations and Systems at the Classical-Quantum Border
1 Introduction
2 Quantum Mechanics and Deformation Theory
3 Quantum Mechanics in the Tomography Approach
4 Applications
4.1 Kinetic Equations and Quantum Corrections
4.2 A Quantum Lyapunov Exponent
5 Quantizers and Dequantizers: An Unified View of Alternative Quantum Formulations
References