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Course |
Syllabus |
Partial differential equations with fractional diffusion
J-M. Roquejoffre (Univ. P.Sabatier. Toulouse III)
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Lect. 1. Introduction and motivations
- Models involving nonlocal diffusions: population dynamics, boundary reactions
- Representation formulas for the fractional Laplacian
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Lect. 2. The dynamics of Fisher-KPP models
- KPP type equations with classical diffusion: travelling waves, linear in time propagation
- KPP type equations with fractional diffusion: exponential in time propagation
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Lect. 3. Free boundary problems for the fractional Laplacian
- Boundary behaviour of a fractional harmonic function
- The one-phase free boundary problem and its weak formulation
- Qualitative properties
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Lect. 4. Nonlocal minimal surfaces (i)
- Motivation: generation of nonlocal motions by convolution with a Poisson kernel
- Minimising fractional Sobolev norms
- Viscosity properties
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Lect. 5. Nonlocal minimal surfaces (ii)
- The de Giorgi theorem for regularity forminimal surfaces
- Extension to the nonlocal case
- Open questions
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Mathematical modelling of phase transitions
T. Myers (CRM, Bellaterra, Barcelona)
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Lect. 1. Introduction to Stefan problems
- Derivation of the heat equation and Stefan condition
- Classical solutions to the heat equation
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Lect. 2. Techniques
- Non-dimensionalisation
- Separation of variables
- Perturbation methods
- Similarity solutions
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Lect. 3. Integral methods for the heat equation
- Laplace transforms
- Heat balance integral methods (HBIM)
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Lect. 4. Solution of Stefan (phase change) problems
- The Neumann solution
- Travelling waves
- Approximate methods (Integral methods, perturbation, boundary fixing ...)
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Lect. 5. Practical applications
- Atmospheric icing (structural/aircraft)
- Contact melting and Leidenfrost
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New connections between dynamical systems and Hamiltonian
PDEs
M. Berti (Univ. Federico II, Napoli)
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Lect. 1. Introduction to Hamiltonian PDEs. Examples: nonlinear schrodinger
and wave equation, KdV equation, etc... as infinite dimensional
Hamiltonian systems
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Lect. 2. The problem of existence periodic and quasi-periodic solutions, the
small divisor problem, history of some known results
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Lect. 3. An abstract Nash-Moser implicit function theorem with parameters
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Lect. 4. Existence of periodic and quasi-periodic solutions for the Nonlinear
Schrodinger equation
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Lect. 5. Estimates for the linearized problem
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Action-minimizing methods in Lagrangian dynamics
A. Sorrentino (Univ. of Cambridge)
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1. In the beginning were KAM tori". Action
minimizing properties of measures and orbits on KAM tori and
invariant Lagrangian graphs: a cartoon example
- Lect.
2. Todo sobre Mather". Action-minimizing
probability measures for Tonelli Lagrangians: existence and
properties. The Mather sets
- Lect.
3. A phase-space Odissey". Action-minimizing
orbits: existence and properties. The Mañé and Aubry sets
- Lect.
4. The usual suspects". Hamilton-Jacobi equation
and Fathi's weak KAM theory
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