Advanced Topics in Partial Differential Equations (winter term 2018/19)
- The first half of the course (8 weeks) will be given by Prof. Zacher, the second by Prof. Wiedemann.
- Both lecturers will briefly describe the main topics of their lectures on Tuesday 16/10/2018.
- Please register for the course in Moodle.
When and where does the course take place?
- Tuesday, 08:15 - 09:45 in room 120 (Helmholtzstraße 18)
- Thursday, 10:15 - 11:45 in room E60 (Helmholtzstraße 18)
- Wednesday, 14:15 - 15:45 in room E60 (Helmholtzstraße 18)
Part I (Zacher)
The first half of the course is concerned with several aspects of regularity theory for elliptic and parabolic PDEs. The topics include:
- De Giorgi-Nash-Moser theory for elliptic equations in divergence form (Boundedness and Hölder regularity of weak solutions and Harnack inequalities)
- De Giorgi-Nash-Moser theory for parabolic equations (at least the main ideas)
- L_p-theory for elliptic equations and links to harmonic analysis, maximal regularity
- Schauder theory (estimates in Hölder spaces) for elliptic problems
Part II (Wiedemann)
The second half of the course deals with the Navier-Stokes equation from fluid dynamics. We will discuss:
- the physical motivation for the equations
- existence theory for weak solutions in three dimensions
- uniqueness theory in two dimensions
- partial regularity (if time permits)
- Gilbarg, Trudinger: Elliptic partial differential equations of second order, Springer
- Han, Lin: Elliptic partial differential equations, Courant Lecture Notes
- Wu, Yin, Wang: Elliptic and parabolic equations, World Scientific
- Giaquinta, Martinazzi: An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs
- Ladyshenskaja, Solonnikov, Uraltseva: Linear and quasi-linear equations of parabolic type
- Robinson, Rodrigo, Sadowski: The three-dimensional Navier-Stokes equations: the classical theory, Cambridge University Press, 2016
- Galdi: An introduction to the Navier-Stokes initial-boundary value problem. Birkhäuser, Basel, 2000
There will be an oral exam at the end of the term.
Lectures: Prof. Dr. Emil Wiedemann, Prof. Dr. Rico Zacher
Exercise course: Ibrokhimbek Akramov