Justification of an Approximation Equation for the Bénard-Marangoni Problem

The Bénard-Marangoni problem is a mathematical model for the description of a temperature dependent fluid flow in very thin liquid layers with a free top surface. The liquid is bounded from below by a horizontal plate of a certain temperature. Above the liquid there is an atmosphere cooler than the bottom plate. There is a purely conducting steady state, where the liquid is at rest. This state is stable as long as the difference between the temperature of the bottom plate and the temperature of the atmosphere is sufficiently small. If the temperature difference surpasses a certain threshold, convection sets in, which is mainly driven by surface tension rather than buoyancy. The onset of convection can be seen as the propagation of a spatially periodic pattern, such that we interpret the BénardMarangoni problem as a pattern forming system. In this thesis we are interested in the behaviour of the system when the purely conducting steady state becomes unstable. From the equations of the Bénard-Marangoni problem we formally derive a Ginzburg-Landau like system of modulation equations, which we use to construct approximate solutions for the full problem. In this thesis we prove an approximation theorem for these modulation equations. That means, we show that the approximate solutions lie close to true solutions of the Bénard-Marangoni problem, at least for a long time. The validity of the Ginzburg-Landau approximation was already shown for a number of pattern forming systems. In case of the Bénard-Marangoni problem, however, we have a spectral situation that does not allow a direct application of the existing approximation proofs. Hence, we first consider a toy problem exhibiting such a kind of spectrum and develop a method for proving an approximation result in this case. Furthermore, the existing approximation proofs were restricted to semilinear problems. However, the equations of the Bénard-Marangoni problem are quasilinear. Therefore, we also develop a method for proving approximation results for quasilinear problems. We then turn back to the Bénard-Marangoni problem. After showing local existence and uniqueness of solutions, we apply our new methods in order to prove the desired approximation result.