Speaker
Description
In this talk, we discuss up to two different aspects of the two-phase Mullins--Sekerka evolution.
In the first part, we investigate the convergence to equilibrium configurations: It is well-known that nearly spherical interfaces given as nearly radial graphs over spheres converge to an equilibrium configuration exponentially fast with a rate $\sim 1/R^3$, where $R$ is the radius of the corresponding equilibrium sphere. For increasingly large radii, this exponential rate deteriorates -- this reflects the fact that the spectral gap of the Laplacian vanishes as the period tends to infinity. We show how finer estimates of the dynamics of the flow along the center manifold of spheres allows for obtaining an algebraic convergence rate that persists even in the infinite-period-transition.
In the second part, which is based on ongoing joint work with Wenhui Shi, we might sketch how mild global-in-time solutions for the fully unbounded flow with initial conditions close to a hyperplane may be obtained.