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.

Lecture:

• Tuesday, 08:15 - 09:45 in room 120 (Helmholtzstraße 18)
• Thursday, 10:15 - 11:45 in room E60 (Helmholtzstraße 18)

Exercise course:

• 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)

Part I:

• 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

Part II:

• 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.