MATHEMATICAL PHYSICS II
Academic year and teacher
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- ARIANNA PASSERINI
- Credits
- 6
- Curriculum
- APPLICATIVO
- Didactic period
- Secondo Semestre
- SSD
- MAT/07
Training objectives
- Aim of the course are modeling skills: how to assemble different phenomena in a theory where the solutions of some equations allow the control of the evolution of those phenomena.This is done through a review, as deep as possible, of the mathematical methods in fluid mechanics, from the dynamical symmetries algebra to functional analysis theorems, simply through the study of their applications to that only physical phenomenon consisting in the heat transport by convective fluxes. The discussion of Bénard's convection in 2D starts from the Eulerian approach to continuum dynamics, then, through the derivation of the Newtonian stress tensor, the non dimensional parameters involved in the conservation laws, and the Boussinesq approximation, one gets a writing with insight of the system of PDE, where the border of the applications is known as well as the physical phenomena it is demanding to describe. At the end, the tools to prove existence and stability (even optimal) of the solutions are given
At the end of the course, the student will have acquired, on one hand, notions and tecniques in the modeling of continuum media, and on the other hand, he will have gained experience in the application of abstract methods in Functional Analysis and Calculus of Variations to mechanical (especially non-linear) problems. Prerequisites
- In order to fruitfully attend the course, the student should have notions in the following areas (in addition to the first-two-years undergraduate teachings):
Mathematical-Physics (only basic/general notions are needed)
Functional Analysis (Banach-, Hilbert-, Sobolev spaces; however more or less extensive recalls will be made during the course when necessary) Course programme
- Review of Continuum Mechanics and balance laws (6 hours). Modeling the stress tensor by invariance (12 hours), Boussinesq approximation, the Bénard problem in 2D, non dimensional units, existence of solutions (16 hours). Linear stability and optimal non linear stability (8 hours).
Since the students are not so many, the number of hours devoted to each subject is related to student's previous skills, curiosity and needs. Didactic methods
- The lectures hopefully take place in the classroom and registrations will be available.
Learning assessment procedures
- The final exam consists in an oral discussion on the subjects of the course
Reference texts
- Lecture notes. Inoltre: Morton E. Gurtin, An Introduction to Continuum Mechanics (for modeling by frame invariance);
Fermi E., Termodinamica; Rosati-Lovitch, Fisica Generale II; Galdi G.P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations (for Functional Analysis applied to Fluid Dynamics); Straughan B., The Energy Method, Stability and Nonlinear Convection (for Calcolus of Variations)
