The Navier-Stokes equations describe the flow of viscous and incompressible Newtonian fluids. They are a fundamental system of partial differential equations from fluid mechanics and the basis for many other models and applications. We will give an introduction to the mathematical analysis of the Navier-Stokes equations starting from the linearized Stokes equations and Navier-Stokes system in a time-independent situation. Afterwards time-dependent problems will be studied. In particular, existence of weak and strong solutions as well as regularity and uniqueness of weak solutions will be studied, which is closely related to one of the Millenium Problems of the Clay institute. In dependence on the time and interests of the audience related topics like non-Newtonian fluids, the Euler equations and related systems of partial differential equations will be discussed.
Here you will find additional information for the short lecture series by Richard Höfer.
The lecture series provides an introduction to the theory of partial
differential equations (PDEs). In the first part we will study classical
solution theories for PDEs. In particular we will discuss some
fundamental equations and examples and show limitations of classical
solution concepts. In the second part of the lecture series an
introduction to the modern theory of PDEs is given, which is based on a
weaker notion of solutions and functional analytic concepts. In
particular we will study elliptic PDEs.
The content of the lecture consists of the following parts:
Die meisten gewöhnlichen Differentialgleichungen lassen sich nicht analytisch lösen. Deswegen ist man für Anwendung meistens darauf angewiesen diese näherungsweise mit geeigneten Algorithmen auf dem Computer zu lösen. Es werden grundlegende numerische Verfahren für die Lösung von gewöhnlichen Differentialgleichungen hergeleitet und mathematisch analysiert.
We will give an introduction to the theory of strongly continuous semigroups, which can be used to solve abstract evolution equations of the formu'(t)+Au(t)= f(t)and nonlinear variants. Here A is a suitable (unbounded) linear operator on a Banach space X and the equation can be seen as a Banach space valued ordinary differential equation. A lot of equation can be reformulated in this form, e.g. parabolic and hyperbolic partial differential equations and also equations from stochastic analysis.We will give a basic introduction to this theory and treat several applications, e.g. to the Navier-Stokes and Schrödinger equations.