Nonlinear Partial Differential Equations - M2 - 8EC

1) Prerequisites:

Courses from the bachelor in mathematics on analysis as well as ordinary (and partial) differential equations are required. The course will use functional analysis and PDE methods and apply them to nonlinear partial differential equations, thus establishing a direct link to applications of the Mastermath courses Functional Analysis and Partial Differential Equations at the M1 level. Students should have attended at least one of these courses, but preferably both. The course also has a direct link to the Mastermath course Dynamical Systems at the M1 level, for instance in the topic of spatial dynamics and through the equations from fluid dynamics as these are naturally derived using flow maps. Attending this course is not required but advantageous. 

 

2) Aim of the course:

Much of the phenomena around us are naturally modelled in the language of nonlinear partial differential equations (PDEs) making them invaluable in many application areas, ranging from the life sciences to chemistry and physics. In the first part of the course (taught by Prof. Dr. van Heijster), we will study pattern formation, such as localised structures and travelling waves, arising naturally in systems of reaction-diffusion-advection equations. These patterns range from close- to-equilibria patterns (i.e., Turing patterns) to far-from-equilibria patterns. The existence, stability and interaction of these patterns will be analysed using various viewpoints (e.g. spatial dynamics, infinite-dimensional dynamical systems) and analytic techniques of the modern theory of nonlinear PDEs. 

 

In the second part of the course (taught by Dr. Gnann), we will treat the Stokes and Navier-Stokes equations and construct weak and strong solutions to the latter. This will rely on spectral methods, embedding theorems for Sobolev spaces, interpolation methods, and compactness arguments. Thus, we will prove existence and uniqueness of global weak and strong solutions to the (time-dependent) Stokes equations, existence of global weak solutions to the Navier-Stokes equations, global uniqueness for the 2D Navier-Stokes equations, and existence and uniqueness of strong solutions to the 3D Navier-Stokes equations for short times or small initial data. A discussion on the third Millennium Prize Problem and recent advances in the field rounds up the second half of the lecture.

 

3) Rules about Homework/Exam: 

  • Homework (40%): a total of 6 homework sets need to be handed in during the semester. The combined average counts for 40% towards your final grade

  • Oral examination in Exam weeks (60%): an oral examination, partly based on your homework sets, will take place during the exam weeks. This counts for 60% towards your final grade. 

  • A student can only pass the course if their scores for both the homework and oral examination are at least 5.0 out of 10.

  • If a student fails to score 50% for their homework, an alternative homeworkset will be offered.  

  • If a student fails to score 50% for their oral examination, an alternative oral examination will be offered (in consultation with the student).

4) Lecture notes/Literature:

The teaching material of this course consists of lecture notes, slides and research articles. These will be provided to the students free of charge.

 

5) Lecturers:

Prof. dr. Peter van Heijster (part 1) and Dr. Manuel Gnann (part 2). Potentially, there will be (a) guest lecturer(s) during one or two of the sessions.