$Helsinki University of Technology, Institute of Mathematics$

Graduate minicourse on partial differential equations

June 4-6, 2007

This symposium will bring together researchers as well as graduate students in the field of hyperbolic partial differential equations. Each of the speakers will give an intensive course consisting of about four lectures. The speakers are:

Schedule (preliminary)

Monday	June 4
10.00-10.45	Holden
11.15-12.00	Feireisl
14.00-14.45	Holden
15.15-16.00	Westdickenberg
16.15-17.00	Rascle

Tuesday	June 5
9.15-10.00	Westdickenberg
10.30-11.15	Rascle
11.30-12.15	Holden
13.45-14.30	Holden
15.00-15.45	Feireisl
16.00-16.45	Westdickenberg

Wednesday	June 6
9.15-10.00	Rascle
10.30-11.15	Feireisl
11.30-12.15	Westdickenberg
13.45-14.30	Feireisl
15.00-15.45	Rascle

All talks in Room U 322 (TKK, Main Building, 3rd floor)

For furher information, please contact the members of the organizing committee: Gustaf Gripenberg, Juha Kinnunen and Stig-Olof Londen.

Abstracts

Eduard Feireisl: (Prague): Mathematical theory of viscous, compressible, and heat conductive fluids

We discuss the basic principles of the mathematical theory of general viscous, compressible, and heat conductive fluids. The principal aim is to develop a rigorous existence theory of the Navier-Stokes-Fourier system without any restriction on the size of initial data and the life span of solutions. The basic framework are distributional solutions belonging to the function spaces of Sobolev type. We briefly discuss the main difficulties of the problem and develop new technique to overcome them. Finally, the strength of the theory is demonstrated through several examples: stabilization of solutions for long time, structural stability, and asymptotic (singular) limits.
The programme of the course may be shortly delineated as follows:

Navier-Stokes-Fourier system, physical background
A concept of variational solutions based on the second law of thermodynamics, definitions and basic properties
Principal problems in proving large data existence
Renormalization, the effective viscous pressure, compensated compactness
Weak sequential stability
Long-time behavior of solutions, the set of equilibria and stability
Singular limits: low Mach and Froude number limits, Boussines and anelastic approximation

References:

E.Feireisl. Dynamics of viscous compressible fluids. Oxford University Press, Oxford, 2003.
O.~A. Ladyzhenskaya. The mathematical theory of viscous incompressible flow. Gordon and Breach, New York, 1969.
P.-L. Lions. Mathematical topics in fluid dynamics, Vol.2, Compressible models. Oxford Science Publication, Oxford, 1998.
R.~emam. Navier-Stokes equations. North-Holland, Amsterdam, 1977.

Helge Holden (Trondheim):

We will give a brief but concise introduction to the mathematical theory of hyperbolic conservation laws, that is, nonlinear partial differential equations of the form $u_t+f(u)_x=0$. The course will give the fundamental properties of scalar equations as well as systems in one dimension.

Michel Rascle (Nice):

I will talk about nonlinear (systems of) scalar conservation laws, of the form: $u_t+f(u)_x=0$. I will in particular describe a class of such equation or systems arising in traffic flow modeling, which turn out to be a kind of intermediate systems (Temple, Keyfitz-Kranzer ..), which allow for both a much easier analysis than general systems and a much richer behaviour that single conservation laws. I will also mention, for scalar conservation laws, some work related to the strong $L^1$ continuity in time of the solution, in particular when the time approaches $0$.

Michael Westdickenberg (Atlanta): Finite Energy Solutions of the Isentropic Euler Equations with Geometric Effects

In this course we consider the system of isentropic Euler equations, which model the dynamics of compressible fluids under the simplifying assumption that the thermodynamical entropy is constant in space and time. We are especially interested in multi-dimensional spherically symmetric flows, which are relevant for several important applications. We will develop an existence theory for the Cauchy problem and show that for any initial data with finite mass and energy, weak solutions to the isentropic Euler equations do exist.