Pattern Formation and Partial Differential Equations

The research I present is motivated by specific, but ubiquitous pattern in models from physics: Domain and wall patterns the magnetization forms in ferromagnets, the coarsening of the phase distribution in demixing of polymer blends, the roughening of a crystal surface under deposition.

Dynamically speaking, the type of models ranges from variational formulations, over (driven) gradient flows to non-gradient systems. The challenge for a rigorous analysis lies in the fact that we are interested in generic behavior of solutions, as expressed by (experimentally and numerically observed)scaling laws, that hold in the limit of large system sizes. We argue that methods from the theory of partial differential equations can be used to provide at least one-sided, optimal bounds on these scaling laws.

Felix Otto, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany