Plenary Lecture

Some Mathematical Problems in the Theory of Boiling Liquids

Professor Ruediger Landes
Department of Mathematics
University of Oklahoma

Abstract: In a new attempt to describe certain states of boiling liquids mathematically, such as “nucleate boiling, transient boiling and pool boiling” Professor W. Marquart from Aachen proposed to consider the heat equation in the wall of the heater subject to a nonlinear Neumann boundary condition towards the boiling liquid, but without any attempt of modeling the boiling liquid itself. The results that we present indicate that this parabolic equation with the nonlinear boundary condition indeed has solutions with properties observed experimentally. Such as the existence of traveling wave solutions which model the phase ransition from nucleate boiling to transient boiling. We also construct weak supersolutions which provide bounds for the propagation speed of such an phase transitions. Further, we are able to construct stable supersolutions to initial configurations with locally supercritical values, indicating that at least for a short time both states may exists together, as it has been observed experimentally. To study this phenomenon further, Speetjens et. al. have conjectured that the solution gives rise to a semigroup with an attractor being the unstable manifold of its fixed point set. We verify that the solution has indeed that property. But we point out that the semigroup should be considered acting on L2 and not on H1 as suggested in their conjecture.

Brief Biography of the Speaker: Rüdiger Landes studied mathematics at University of Munich and the University of Göttingen and received his Diploma from the University of Göttingen. He graduated with an PhD from the University of Bochum and then worked for one Year at the SFB 272 in Bonn. In 1984 he promoted with a Habilitation at the University of Bayreuth. Thereafter he spend a year on visiting appointment at the University of Chicago and then joined the faculty at the University of Oklahoma. His research is devoted to nonlinear problems in partial differential equations. The main areas of his work are: The existence of solution elliptic and parabolic equations and systems with perturbations of unlimited growth, the existence, uniqueness and regularity of quasilinear elliptic systems with critical growth, and the dynamical behavior of solutions of parabolic problems with nonlinear boundary conditions. He was on research sojourns at the University of Oulu, Finland, at the University of Catania, in Italy, with SFBs at University Bonn and at the University Heidelberg, Germany twice, at the MSRI, Berkely, USA, and at University Dresden in Germany. He is member of the editorial board of Applied and Abstract Analysis and the International Journal of Pure Mathematics.