Mathematical Fluid-Mechanics
A.Y. 2025/2026
Learning objectives
The aim of the course is to provide an overview of the modern mathematical theory in fluid mechanics from the theory of Kato for local and global existence of Euler and Navier-Stokes equations till the analysis of some "small divisors problems" like the construction of periodic and multi periodic nonlinear waves in Euler and Water Waves equations.
Expected learning outcomes
The student will be aware of the energy methods to study local and global existence of Euler and Navier-Stokes equations in 2D. Moreover the student will be aware of some perturbative methods (typical of mathematical physics) to construct periodic and multi-periodic traveling waves in fluid mechanics, based on Normal Forms, KAM and Nash-Moser methods and tools from micro-local analysis.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
The aim of the course is to provide an introduction to the modern theory of Mathematical Fluid mechanics with a particular focus on the analysis of the Euler, Navier-Stokes and Water Waves equations (and some of their approximate models). I will start by presenting the classical results till the modern frontiers of the research. A short sketch of the syllabus is provided below.
Physical introduction to fluids and derivations of Euler, Navier Stokes and Water Waves equations (and some of their approximate models).
Euler and Navier Stokes equations. Analysis of the solutions and long time behavior: local existence, well posedness and vanishing viscosity Kato's method. Global existence and large time behavior for bi-dimensional fluids: Beale-Kato-Majda criterion.
Small divisors problems and construction of nonlinear waves in fluid mechanics. Periodic and multi-periodic traveling waves: Nash-Moser methods and perturbative techniques based on micro-local analysis.
Physical introduction to fluids and derivations of Euler, Navier Stokes and Water Waves equations (and some of their approximate models).
Euler and Navier Stokes equations. Analysis of the solutions and long time behavior: local existence, well posedness and vanishing viscosity Kato's method. Global existence and large time behavior for bi-dimensional fluids: Beale-Kato-Majda criterion.
Small divisors problems and construction of nonlinear waves in fluid mechanics. Periodic and multi-periodic traveling waves: Nash-Moser methods and perturbative techniques based on micro-local analysis.
Prerequisites for admission
Basic knowledges of Mechanics, Elementary Analysis, Measure theory, Fourier series and Fourier transform, linear Partial differential equations.
It is also necessary to attend the course of Real Analysis.
It is also necessary to attend the course of Real Analysis.
Teaching methods
LECTURES
Teaching Resources
LECTURE NOTES OF THE COURSE. It is also useful to look at the following monographs.
1) MAJDA, BERTOZZI. "Vorticity and incompressible flow."
Cambridge texts in applied mathematics 2002
2) BEDROSSIAN, VICOL. "The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations: An Introduction."
American Mathematical Society. Graduate studies in Mathematics 225.
1) MAJDA, BERTOZZI. "Vorticity and incompressible flow."
Cambridge texts in applied mathematics 2002
2) BEDROSSIAN, VICOL. "The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations: An Introduction."
American Mathematical Society. Graduate studies in Mathematics 225.
Assessment methods and Criteria
ORAL EXAM
MAT/07 - MATHEMATICAL PHYSICS - University credits: 6
Lessons: 42 hours
Professor:
Montalto Riccardo
Professor(s)
Reception:
Wednesday, 13.30-17.30
Room 1005, Department of Mathematics, Via Saldini 50, 20133, Milan