Surface diffusion (SD) is a curvature driven flow where a (hyper-)surface evolves by the surface Laplacian of its mean curvature. It is a fourth order parabolic equation. Compared with its second order counterpart, motion by mean curvature (MMC), for which maximum principle is available, much less is known for SD. I will present a stability result for self-similarity solutions. Though the approach is based on linearization, and it only works for the evolution of graphs, it is quite robust and works for both MMC and SD. I will also present an attempt to analyze the pinch-off phenomena for axisymmetric surfaces. The former is based on a joint work with Hengrong Du and the later with Gavin Glenn.

Zoom

Propagation has been modelled by reaction-diffusion equations since the pioneering works of Fisher and Kolmogorov-Peterovski-Piskunov (KPP). Much new developments have been achieved in the past a few decades on the modelling of propagation, with traveling wave and related solutions playing a central role. In this talk, I will report some recent results obtained with several collaborators on some reaction-diffusion models with free boundary and "nonlocal diffusion", which include the Fisher-KPP equation (with free boundary) and two epidemic models.  A key feature of these problems is that the propagation may or may not be determined by traveling wave solutions.  There is a threshold condition on the kernel functions which determines whether the propagation has a finite speed or infinite speed (known as accelerated spreading). For some typical kernel functions, we obtain sharp estimates of the spreading speed (whether finite or infinite).

Zoom

The purpose of this presentation is to introduce a comprehensive variational principle that allows one to apply critical point theory on closed proper subsets of a given Banach space and yet, to obtain critical points with respect to the whole space. This variational principle has many applications in partial differential equations while unifying and generalizing several results in nonlinear Analysis such as the fixed point theory, critical point theory on convex sets and the principle of symmetric criticality.

Zoom

We consider a model for collective behaviour with intrinsic interactions on Riemannian manifolds. We establish the well-posedness of measure solutions, defined via optimal mass transport, on several specific manifolds (sphere, hypercylinder, rotation group SO(3)), and investigate the mean-field particle approximation. We study the long-time behaviour of solutions, where the primary goal is to establish sufficient conditions for a consensus state to form asymptotically. The analytical results are illustrated with numerical experiments that exhibit various asymptotic patterns.

Zoom link

In recent years, there has been significant interest in continuum models of crystal surface evolution and facet formation. However, in the most physically relevant case, when the free energy of the surface is the total variation energy, even existence of solutions to the continuum PDE is unknown. Furthermore, attempts at developing a robust numerical method for simulating solutions suffer from significant stiffness, preventing numerical study of the equation’s behavior on fine spatial grids. In this talk, I will describe a new approach to simulating solutions of the crystal surface evolution equation based on combining the formal gradient flow structure of this equation with modern operator splitting techniques. This is based on joint work with Jian-Guo Liu, Jianfeng Lu, Jeremy L. Marzuola, and Li Wang.

Zoom link