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Differential Equations in Theory and Applications

Spring term 2010

The topic is differential equations (both partial and ordinary differential equations).

Franz Fuchs Why is the corona of the Sun so hot?
Abstract: We simulate and study wave propagation in stellar atmospheres, in particular the atmosphere of the Sun. One of the mysteries to solve is the so called coronal heating problem. It relates to the question of why the temperature of the Sun's corona is O(10^3) times higher than that of the surface. The heating of the atmosphere seems to violate the second law of thermodynamics. This phenomenon is only just now becoming understood, using data from modern solar observation satellites, and particularly from the results of numerical modeling. There are two important energy transport mechanisms in the solar atmosphere; convection generated waves and radiation.

Our model is a combination of the M1-model of radiative transfer and the MHD equations. The equations of MHD model the processes in the photosphere and chromosphere fairly well. But in order to have an appropriate model for the corona one has to account for the effects of radiation. Even though the resulting system is hyperbolic, the design of numerical schemes is hindered by the fact that the maximum eigenvalues of the M1 model are of the order of the speed of light and therefore waves move typically O(104) times faster than the fastest waves in MHD. In order to be able to do any serious numerical simulations of wave propagation with this model, we devise appropriate semi-implicit schemes that can handle those huge differences in time scales.

We present our results comparing explicit and semi-implicit numerical schemes. The new semi-implicit schemes are 4 orders of magnitude more efficient than explicit schemes. Furthermore, we present what effect radiation has on wave propagation. This investigation clarifies the importance of the effects of radiation in the corona and is vital for the understanding of the processes in stellar atmospheres in general and of the Sun's corona in particular.

Aud. F3
Erik Lindgren Obstacle-type problems
Abstract: I will introduce the classical obstacle problem and discuss what properties usually interest people. The main difficulties and singularities that occur will also be discussed. Some brief ideas of proofs will also be given. From there I will go on to the similar “two-phase obstacle problem”. The major differences to the obstacle problem will be discussed and also how these can be dealt with in the proofs.
Room 734 (SB2)
Emmanuel Chasseigne Large Deviations Estimates for Levy-type non-local equations.
Abstract: We study large deviations for some non-local parabolic type equations. We show that, under some assumptions on the non-local term, problems defined in a bounded domain converge with an exponential rate to the solution of the problem defined in the whole space. We compute this rate in different examples, with different kernels defining the non-local term, and it turns out that the estimate of convergence depends strongly on the decay at infinity of that kernel.
Room 734 (SB2)
Trygve Karper The compressible Stokes equations
Abstract: In the 1930s J. Leray proved the existence of solutions to the incompressible Navier–Stokes equations. A comparable theory for the compressible Navier–Stokes equations was established only recently (1990s). The existence theory is due to P-L. Lions and it was a large contributing factor for awarding him the Fields medal.

In this lecture, I will give an easy introduction to the existence theory of P-L. Lions. To introduce the ideas and techniques I will begin by proving existence of solutions to the simpler compressible Stokes equations.

Keywords are: weak convergence methods, compensated compactness, renormalized solutions, and convexity arguments.

Room 734 (SB2)
Trygve Karper The compressible Stokes equations, part II
V. Julin
Convexity criteria and uniqueness of absolutely minimizing functionals
CANCELLED due to force majeure
Teemu Lukkari The porous medium equation with measure data
Abstract: We discuss proving the existence of solutions to the porous medium equation with a “bad” source term. More precisely, we deal with right hand sides given by positive, finite Borel measures; for instance, Dirac's delta is allowed. The proof relies on a priori estimates, and a compactness argument where the structure characteristic of the porous medium equation needs to be taken into account.


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