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Topics in Mathematical Fluid Dynamics

  • Alexander Shnirelman
E2 10 (Leon-Lichtenstein)

Abstract

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The Ideal Incompressible Fluid is the most fundamental model of a continuous media. In this model, the configuration space of the fluid is the group D of volume- preserving diffeomorphisms of the flow domain M. D is an infinite-dimensonal manifold endowed with a weak Riemannian metric (the kinetic energy). The fluid flows (in the absence of the external forces) are the geodesics in this metric. There are two basic problems about the geodesics:
  1. Find the geodesic trajectory for the given initial fluid configuration and velocity (the initial-value problem);
  2. Find the geodesic for given fluid configurations at two different time moments, say at t=0 and t=1 (the 2-point problem).

In this course we consider both problems.

The course includes the following topics:

  1. 2-point problem in 2-dimensional domain. The variational approach and its failure. Genaralized flows and generalized braids. Weak solutions of the variational problem, their regularity.
  2. The category of quasi-ruled manifolds and quasi-ruled maps.
  3. The quasi-ruled structure of the manifold D. Its possible significance for the solvability of the 2-point problem.
  4. The geometrical nature of the exponential map Exp on D: Exp is a qiasi-ruled map.
  5. The analytical nature of the geodesic exponential map Exp on the manifold D : Exp is an elliptic paradifferential operator of order zero.
  6. Open problems.

Keywords
Group of diffeomorphisms, Euler equations, braids, variational methods, generalized flows and braids, quasiruled manifolds and maps, paradifferential calculus

Prerequisites
just the basics of Functional Analysis, Riemannian Geometry, and Measure Theory

Audience
graduate students and above

lecture
01.04.24 31.07.24

Regular lectures Summer semester 2024

MPI for Mathematics in the Sciences / University of Leipzig see the lecture detail pages

Katharina Matschke

MPI for Mathematics in the Sciences Contact via Mail

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