Evolution Equations
- Professors:
- Veneroni Marco, Segatti Antonio
- Year:
- 2017/2018
- Course code:
- 500699
- ECTS:
- 6
- SSD:
- MAT/05
- DM:
- 270/04
- Lessons:
- 48
- Period:
- II semester
- Language:
- Italiano
Objectives
By studying several important models, this course will provide fundamental tools for the analysis and the understanding of evolution partial differential equations.
Teaching methods
Lectures and exercise sessions
Examination
Oral exam
Prerequisites
Basic concepts of functional analysis, Lebesgue integration theory and Sobolev spaces (the main results will be recalled during the course).
Syllabus
-Heat equation (linear diffusion): existence theory for the Cauchy and the Dirichlet problems and qualitative properties of the solution.
-Porous medium equation (nonlinear diffusion): Different solution concepts: classical, strong, and weak solutions. Existence theory for the Cauchy and the Dirichlet problems.
-Gradient flows. Classical theory in Hilbert spaces; an introduction to gradient flows in metric spaces. An application to the Fokker-Planck equation.
-Systems of conservation laws. Integral solutions, hyperbolic systems. The Riemann problem, rarefaction and shock waves. Entropy criteria, vanishing viscosity, entropy/entropy flux pairs.
-Porous medium equation (nonlinear diffusion): Different solution concepts: classical, strong, and weak solutions. Existence theory for the Cauchy and the Dirichlet problems.
-Gradient flows. Classical theory in Hilbert spaces; an introduction to gradient flows in metric spaces. An application to the Fokker-Planck equation.
-Systems of conservation laws. Integral solutions, hyperbolic systems. The Riemann problem, rarefaction and shock waves. Entropy criteria, vanishing viscosity, entropy/entropy flux pairs.
Bibliography
-L.C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, 2002.
-L.C. Evans, Weak convergence methods for Nonlinear Partial Differential Equations, CBMS Regional Conference Series in Mathematics, 74. Published by the American Mathematical Society, 1990.
-R. Jordan, D. Kinderlehrer, and F. Otto. The Variational Formulation of the Fokker- Planck Equation. SIAM Journal on Mathematical Analysis, 29(1):1–17, 1998.
-J.L. Vazquez, Porous Medium Equations. Mathematical Theory. Oxford University Press 2006.
-L.C. Evans, Weak convergence methods for Nonlinear Partial Differential Equations, CBMS Regional Conference Series in Mathematics, 74. Published by the American Mathematical Society, 1990.
-R. Jordan, D. Kinderlehrer, and F. Otto. The Variational Formulation of the Fokker- Planck Equation. SIAM Journal on Mathematical Analysis, 29(1):1–17, 1998.
-J.L. Vazquez, Porous Medium Equations. Mathematical Theory. Oxford University Press 2006.
Modules
- Professor:
- Veneroni Marco
- Lessons:
- 24
- ECTS:
- 3
- Professor:
- Segatti Antonio
- Lessons:
- 24
- ECTS:
- 3
Dipartimento di Matematica ''F. Casorati''
Università degli Studi di Pavia -
Via Ferrata, 5 - 27100 Pavia
Tel +39.0382.985600 - Fax +39.0382.985602