Continuum Mechanics
A.Y. 2023/2024
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 cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
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 and pseudo-differential operators
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 and pseudo-differential operators
Prerequisites for admission
Basic knowledges of Mechanics, Elementary Analysis, Measure theory, Fourier series and Fourier transform, linear Partial differential equations
Teaching methods
LECTURES IN PERSON
Teaching Resources
LECTURE NOTES OF THE COURSE. It would be also useful to look at the following books
1) STOKER. "Water Waves: the mathematical theory with applications."
Wiley classic library edition. 1992
2) WHITHAM. "Linear and nonlinear waves. "
John Wiley and sons
3) MAJDA, BERTOZZI. "Vorticity and incompressible flow."
Cambridge texts in applied mathematics 2002
4) BEDROSSIAN, VICOL. "The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations: An Introduction."
American Mathematical Society. Graduate studies in Mathematics 225.
5) LANNES. "The water waves problem: mathematical analysis and asymptotics."
Mathematical surveys and monographs, volume 188. American Mathematical Society.
1) STOKER. "Water Waves: the mathematical theory with applications."
Wiley classic library edition. 1992
2) WHITHAM. "Linear and nonlinear waves. "
John Wiley and sons
3) MAJDA, BERTOZZI. "Vorticity and incompressible flow."
Cambridge texts in applied mathematics 2002
4) BEDROSSIAN, VICOL. "The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations: An Introduction."
American Mathematical Society. Graduate studies in Mathematics 225.
5) LANNES. "The water waves problem: mathematical analysis and asymptotics."
Mathematical surveys and monographs, volume 188. American Mathematical Society.
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