[Dottorcomp] Fwd: PhD course "Mathematical Aspects of Fluid Mechanics"

[Frediani Paola]

Buongiorno,

inoltro questo annuncio di un corso di dottorato al Politecnico di Milano.
Gli interessati sono invitati a contattare il prof. Andrea Giorgini.

Cordiali saluti,

Paola Frediani

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Course title: Mathematical Aspects of Fluid Mechanics

Lecturer: Andrea Giorgini

Location: Department of Mathematics, Politecnico di Milano

Subject: Introduction to the initial-boundary value problems for the
Navier-Stokes and Euler equations for incompressible fluids.

Goal: The aim of the PhD course is to provide an introduction to the
classical theoretical results concerning the incompressible Navier-Stokes
and Euler equations in both two and three dimensional domains. This course
is meant for a broad audience of PhD students interested in the analysis of
evolutionary PDEs with focus on Fluid Mechanics. Prerequisite topics are
Sobolev spaces and variational methods for linear PDEs (e.g. Lax-Milgram
theorem).

Programme of the course: Sobolev spaces of soleinodal functions,
interpolation inequalities in two and three dimensions, Helmholtz-Weyl
decomposition, De Rham theorem, the steady Stokes problem: existence,
uniqueness and regularity, notion of Leray-Hopf (LH) weak solution to the
Navier-Stokes (NS) equations, global existence of LH weak solutions and
existence of the pressure in two and three dimensional domains, uniqueness
of LH weak solutions in two dimensions, conditional uniqueness of LH weak
solutions in three dimensions, bounds on weak solutions a là
Foias-Guillopé-Temam, Serrin’s conditions for regularity of LH weak
solutions in three dimensions, additional regularity for LH weak solutions
in two dimensions a là Chemin-Lerner and He, existence of global/local
strong solutions to the NS equations in two and three dimensional domains,
large times stabilization of the LH weak solutions in absence of external
force in three dimensions, Lagrangian representation of the NS flow, notion
of weak and strong solutions to the Euler equations in two and three
dimensions, existence of local solutions in Hölder and Sobolev spaces,
Beale-Kato-Majda criteria for breakdown of smooth solutions to the Euler
equations, global well-posedness of the Euler equations in two dimensions
(Yudovich theory), small scale creation and growth of Sobolev norms for the
Euler equations in two dimensions.

Total number of hours: 27h.

Schedule: The course will be held through in-person weekly meetings of 3
hours each from January to March 2024.

Evaluation: The final evaluation will consist of an oral exam on the
covered material or the presentation of a research paper proposed by the
lecturer.
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