The Cahn-Hilliard equation, which models the concentration dynamics of binary alloys, has been object of study for the physics and mathematics community for decades. While existence and uniqueness properties are well-studied and established, certain „geometric“ properties of its solution, in particular their rigorous justification, are still mostly out of reach. This includes growth bounds on the solution's supremum norm and the fact that the dynamics shows a characteristic pattern size.
This dissertation, submitted by the author as a conclusion of his PhD research efforts, presents an approach for each of those two properties. We present an intuitive way of estimating the magnitude of the solution of the Cahn-Hilliard equation for a given lengthscale parameter. The main idea is that (spatial) local extrema of the solution are approximately normally distributed with an standard deviation independent from the parameter. As the number of local extrema grows linearly, the supremum norm corresponds asymptotically to a maximum of i.i.d. Gaussians. Then, standard theory about the distribution of the maximum of an ensemble of Gaussians yields a logarithmic growth. The second part combines ideas from probability theory and ergodic theory with a unique idea by Edelman and Kostlan to obtain an estimate on the pattern size of the equation's mild solution. Essentially, we derive ergodic averaging statements about weighted sequences on arbitrary summation domains to obtain an asymptotic expression for an integral over a highfrequency function. The theory is backed up with numerical simulations using the Octave language interspersed throughout the text.
Dissertation zur Erlangung des akademischen Grades eines Dr. rer. nat., eingereicht an der Mathematisch-Naturwissenschaftlich-Technischen Fakultat der Universitat Augsburg von Philipp Wacker, Augsburg, Mai 2016
Geometric properties of mild solutions of the stochastic Cahn-Hilliard equation
Augsburg 2016. 146 Seiten. Paperback.