Consider a diffusive passive scalar advected by a two
dimensional incompressible flow. If the flow is cellular (i.e.\ has a
periodic Hamiltonian with no unbounded trajectories), then classical
homogenization results show that the long time behaviour is an effective
Brownian motion. We show that on intermediate time scales, the effective
behaviour is instead a fractional kinetic process. At the PDE level this
means that while the long time scaling limit is the heat equation, the
intermediate time scaling limit is a time fractional heat equation. We
will also describe the expected intermediate behaviour in the presence
of open channels.
In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of the Allen-Cahn equation. By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers.
This is a joint work with H. Ninomiya, K. Todoroki.
The general average distance problem, introduced by Buttazzo, Oudet, and Stepanov, asks to find a good way to approximate a high-dimensional object, represented as a measure, by a one-dimensional object. We will discuss two variants of the problem: one where the one-dimensional object is a measure with connected one-dimensional support and one where it is an embedded curve. We will present examples that show that even if the data measure is smooth the nonlocality of the functional can cause the minimizers to have corners. Nevertheless the curvature of the minimizer can be considered as a measure. We will discuss a priori estimates on the total curvature and ways to obtain information on topological complexity of the minimizers. We will furthermore discuss functionals that take the transport along the network into account and model best ways to design transportation networks. (Based on joint works with Xin Yang Lu and Slav Kirov.)
This is a joint work with Piermarco Cannarsa and Wei Cheng.
We study the properties of the set S of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation, for a Tonelli Hamiltonian.
The main surprise is the fact that this set is locally arc connected—it is even locally contractible. This last property is far from generic in the class of semi-concave functions.
We also “identify” the connected components of this set S.
This work relies on the idea of Cannarsa and Cheng to use the positive Lax-Oleinik operator to construct a global propagation of singularities (without necessarily obtaining uniqueness of the propagation).
High-dimensional data can be difficult to analyze. Assume data are distributed on a low-dimensional manifold. The Vector Diffusion Mapping (VDM), introduced by Singer-Wu, is a non-linear dimension reduction technique and is shown robust to noise. It has applications in cryo-electron microscopy and image denoising and has potential application in time-frequency analysis.
In this talk, I will present a theoretical analysis of the effectiveness of the VDM. Specifically, I will discuss parametrisation of the manifold and an embedding which is equivalent to the truncated VDM. In the differential geometry language, I use eigen-vector fields of the connection Laplacian operator to construct local coordinate charts that depend only on geometric properties of the manifold. Next, I use the coordinate charts to embed the entire manifold into a finite-dimensional Euclidean space. The proof of the results relies on solving the elliptic system and provide estimates for eigenvector fields and the heat kernel and their gradients.