Exercise set 1
Exercise set 2
Exercise set 3 (exercise 5 corrected)
Exercise set 4 (exercise 4 corrected)
Exercise set 5
|Measure and Integral (MAT21007), Real analysis I (MAST31001), Functional analysis (MAST31002), Navier-Stokes equations I (MAST31042).|
|The simultaneous course Sobolev Spaces (MAST31016) is very useful; we use basic properties of Sobolev spaces in the course.|
We discuss the Millennium Problem about the existence of smooth solutions of the Navier-Stokes equations. We go through a proof of a partial result, the existence of weak (Leray-Hopf) solutions. Along the way, we develop a lot of mathematical machinery that is not specific to Navier-Stokes equations but also vital in the study of mathematical fluid dynamics in general
Navier-Stokes equations are a set of partial differential equations (PDE’s) that describes e.g. the flow of water. The equations were introduced by C.-L. Navier almost two centuries ago, but the fundamental question about existence of smooth solutions (for every smooth initial data) is still open. It has been chosen as one of the seven Millennium Problems of the Clay Mathematics Institute.
We next briefly describe the modern way to attack the problem. Given a smooth initial data, it is very difficult to construct a smooth solution directly. Therefore, one first attempts to prove the existence of a weak solution (which need not be differentiable in the usual sense). Once that is achieved, one hopes to prove that the weak solution is smooth (and unique).
- Leray and E. Hopf have proved the existence of weak solutions of the Navier-Stokes equations (even for non-smooth square integrable initial datas), and their solutions nowadays bear their names. Besides the Millennium Problem, another famous open question is whether for every square integrable initial data, the Leray-Hopf solution is unique.
The main goal of the course is to give a proof of the existence of Leray-Hopf solutions. In the course of the proof, we introduce many central tools of modern mathematical fluid dynamics such as Bochner spaces, the Helmholtz-Hodge decomposition, Galerkin approximation, the Rellich Compactness Theorem and the Aubin-Lions Lemma. If time permits, we discuss some known results on the two open problems mentioned above. Unlike in the course Navier-Stokes equations I (MAST31042), we use Lebesgue integrals and functional analysis.
Useful related textbook include e.g.:
- J. C. Robinson, J. L. Rodrigo and W. Sadowski: The Three-Dimensional Navier-Stokes Equations (2016)
- R. Temam: Navier-Stokes equations (1977)
- A. Bertozzi and A. Majda: Vorticity and Incompressible Flows (2002)
- C. R: Doering and J. D. Gibbon: Applied analysis of the Navier-Stokes equations (1995)
- G. P. Galdi: An introduction to the mathematical theory of Navier–Stokes equations. Steady state problems. Second edition (2011)
- H. Sohr: The Navier–Stokes equations: an elementary functional analytic approach (2001)
|Lectures and exercise classes|
|Exam and exercises, course will be graded with grades 1-5|