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Session 23. Nonlinear Evolution Equations and their Applications
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Moving boundary problems describing osmosis: Modelling, well-posedness, stability |
Georg Prokert, TU Eindhoven, Faculty of Mathematics and Computer Science, The Netherlands
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The talk is based on joint work with F. Lippoth, Leibniz Universität Hannover,
and M.A. Peletier, TU Eindhoven.
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We discuss a moving boundary problem describing the motion of a
closed, semi-permeable membrane separating two phases of a salt
solution. The motion is driven by osmotic pressure and surface
tension. By transformation to a fixed reference domain, the problem
takes the form of a fully nonlinear, coupled system of evolution
equations inside the domain and on its boundary. Its linearization
is a parabolic system with boundary conditions of relaxation type in
the sense of Denk, Prüss, and Zacher. By application of
corresponding maximal regularity results, we show the existence of
classical solutions for short time and, using spectral analysis,
global existence and normal stability of the manifold of stationary
solutions.
The problem has a variational structure as generalized gradient flow
with the sum of entropy and surface area playing the role of free
energy. We use this idea to generalize our model to include slow
viscous flow of the solvent.
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