Transport PDEs in Kinetic Theory and Fluid Mechanics
Additional Info
- ECTS credits: 6
- University: University of Granada
- Semester: 3
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Topics:
1 Transport Models:
Conservation laws (fluids, traffic, etc.) and kinetic models (Liouville, Vlasov, Boltzmann equations).
2 Linear Transport Equations:
Initial value problems, first-order equations with regular and singular fields, characteristic equations, and associated dynamical systems.
3 Introduction to Nonlinear Scalar Conservation Laws:
Riemann-Hugoniot conditions and admissibility criteria for singularities.
4 Introduction to Fluid Mechanics Equations:
Fundamental equations governing fluid dynamics.
5 The Liouville Equation in Kinetic Theory:
Derived models: free transport equation, Vlasov-Poisson systems, Vlasov-Maxwell equations, and Boltzmann and Vlasov-Poisson-Fokker-Planck equations.
6 Generalities on the Vlasov-Poisson System:
Invariances and conserved quantities, a priori estimates, moment control, weak formulation, moment lemmas, and existence results. Asymptotic behavior in the repulsive case: pseudoconformal law.
7 Orbital Stability of Galaxies:
Scattering in gravitational systems and study of polytropes.
8 Study of Coupled Vlasov-Maxwell Models:
Analysis of relativistic kinetics and the interaction between charged particles and electromagnetic fields.
