Conference Agenda

The Lojasiewicz-Simon inequality is a powerful tool to establish long time existence and convergence of solution to a gradient flow starting sufficiently near to a local minimum of the energy. In this talk we present a new version of this inequality for functionals on infinite dimensional manifolds. This can be used, for instance, to prove stability of a Willmore free boundary problem.

In the higher dimensional setting, critical mass phenomena are known to occur in the quasilinear Keller--Segel system for a variety of different diffusion rates $D(u)$ and taxis sensitivity functions $S(u)$ being critical in the sense that $S(u)/D(u) \sim u^{n/2}$ for large $u$, where $n$ denotes the space dimension. The most famous example is the two-dimensional minimal Keller--Segel system given by $D(u)=1$ and $S(u)=u$, for which the mass $4\pi$ (or $8\pi$ in the radially symmetric case) distinguishes between boundedness and the possibility of blow-up.

In this talk, based on a recent joint work with Xinru Cao, it is shown that this is no longer the case for one-dimensional domains: Solutions of the quasilinear system with $D(u)=(u+1)^{m−1}$ and $S(u)=u(u+1)^m$ for (many) $m \in \mathbb R$ emanating from initial data with arbitrary large mass are globally bounded. Accordingly, the absence of a critical mass phenomenon appears to be a general property of the one-dimensional setting and is not limited to the case $m=0$ already studied in the literature.

The coupling of the evolution of a surface with evolution equations defined on that surface is of relevance in many applications and has been in the focus of interest in the analysis of parabolic PDEs in recent years. In applications the evolution of two-phase vesicles and biomembranes is governed by flows decreasing an energy which involves Canham-Helfrich-type curvature energies coupled to a Ginzburg-Landau energy. We derive a new Cahn-Hilliard-Canham-Helfrich system for the evolution of two-phase membranes. The resulting system is highly non-linear and we use the theory of quasi-linear parabolic evolution equations in weighted $L_p$-spaces to show the existence of a strong local-in-time solution and hence demonstrate that the derived system is well-posed. This is a joint work with Harald Garcke (Regensburg).

We study several models for the interaction of a fluid with a free surface, where the momentum balance on the surface incorporates an elastic stress.