We examine a nonlocal porous medium equation, $\partial_t u + \operatorname{div}(u \nabla L^{-1} u) = 0$, where the pressure is governed by a general symmetric Lévy operator, $L$, which is widely recognized as the generator of a symmetric Lévy stochastic process. Namely, we consider a nonlocal operator of the form \begin{equation} Lu(x)=\operatorname{p.v.}\int_{\mathbb{R}^d} (u(x)-u(y))\nu(x-y)\mathrm{d} y (x\in\mathbb{R}^d), \end{equation} where $\min(1,|h|^2)\nu\in L^1(\mathbb{R}^d)$ and $\nu(h)=\nu(-h)$. A nonlocal symmetric Lévy operator generalizes the classical fractional Laplace operator $(-\Delta)^s $, with $s \in (0,1) $. We construct weak solutions in the context of the corresponding nonlocal Sobolev space using the Jordan-Kinderlehrer-Otto (JKO) minimizing movement scheme. The absence of interpolation and various tools from classical fractional Sobolev spaces renders our approach more challenging. Furthermore, we shall investigate the nonlocal-to-local convergence of the problem.
The MBO or thresholding scheme by Merriman, Bence and Osher is one of the most important approximation schemes for mean curvature flow.
In this talk, we will explain a possible extension of this scheme to level set mean curvature flow on a discrete sampled domain.
In this scheme, the function is evolved by a median filter, i.e., iterated applications of a local median. This results in a movement of the level sets according to mean curvature flow.
We prove convergence of the evolution to the viscosity solution in the limit of vanishing time-step size and growing sample size.
In our work we rigorously derive a limiting model for thin rods, starting from a full 3D model for finite plasticity. We are interested in a scaling of the elastic and plastic energy contributions like h^(-4), where h denotes the thickness of the rod. In the limit this results in a 1D bending theory.
For the derivation we lean on the framework of evolutionary Gamma-convergence for rate-independent systems, introduced by Mielke, Roubíček and Stefanelli in 2008. The main difficulty here is to establish a mutual recovery sequence for the stored energy and dissipation. Strategies have been developed by various authors in order to construct such a sequence, e.g. for linearization or in the von Kármán regime. However, these rely on considering infinitesimal deformations in the limit, which we cannot expect in the bending regime. Our approach relies on a construction based on a multiplicative decomposition of the rotation fields obtained via the rigidity estimate from Friesecke, James and Müller. In order to achieve enough regularity, we consider strain gradient terms in the energy, which act on the two parts of the polar decomposition individually. These terms vanish in the limit.
This is joined work with Stefan Neukamm.
