Large traveling capillary-gravity waves for Darcy flow

We study capillary-gravity and capillary surface waves for fluid flows governed by Darcy's law. This includes flows in vertical Hele-Shaw cells and in porous media (the one-phase Muskat problem) with finite or infinite depth. The free boundary is acted upon by an external pressure posited to be in traveling wave form with an arbitrary periodic profile and an amplitude parameter. For any given wave speed, we first prove that there exists a unique local curve of small periodic traveling waves corresponding to small values of the parameter. Then we prove that as the parameter increases but could possibly be bounded, the curve belongs to a connected set $\mathcal{C}$ of traveling waves. The set $\mathcal{C}$ contains traveling waves that either have arbitrarily large gradients or are arbitrarily close to the rigid bottom in the finite depth case. To the best of our knowledge, this is the first construction of large traveling surface waves for a viscous free boundary problem.Comment: 25 page

Traveling wave solutions to the one-phase Muskat problem: existence and stability

We study the Muskat problem for one fluid in arbitrary dimension, bounded below by a flat bed and above by a free boundary given as a graph. In addition to a fixed uniform gravitational field, the fluid is acted upon by a generic force field in the bulk and an external pressure on the free boundary, both of which are posited to be in traveling wave form. We prove that for sufficiently small force and pressure data in Sobolev spaces, there exists a locally unique traveling wave solution in Sobolev-type spaces. The free boundary of the traveling wave solutions is either periodic or asymptotically flat at spatial infinity. Moreover, we prove that small periodic traveling wave solutions induced by external pressure only are asymptotically stable. These results provide the first class of nontrivial stable solutions for the problem.Comment: Funding source for H. Q. N. correcte

Enhancing Few-shot Image Classification with Cosine Transformer

This paper addresses the few-shot image classification problem, where the classification task is performed on unlabeled query samples given a small amount of labeled support samples only. One major challenge of the few-shot learning problem is the large variety of object visual appearances that prevents the support samples to represent that object comprehensively. This might result in a significant difference between support and query samples, therefore undermining the performance of few-shot algorithms. In this paper, we tackle the problem by proposing Few-shot Cosine Transformer (FS-CT), where the relational map between supports and queries is effectively obtained for the few-shot tasks. The FS-CT consists of two parts, a learnable prototypical embedding network to obtain categorical representations from support samples with hard cases, and a transformer encoder to effectively achieve the relational map from two different support and query samples. We introduce Cosine Attention, a more robust and stable attention module that enhances the transformer module significantly and therefore improves FS-CT performance from 5% to over 20% in accuracy compared to the default scaled dot-product mechanism. Our method performs competitive results in mini-ImageNet, CUB-200, and CIFAR-FS on 1-shot learning and 5-shot learning tasks across backbones and few-shot configurations. We also developed a custom few-shot dataset for Yoga pose recognition to demonstrate the potential of our algorithm for practical application. Our FS-CT with cosine attention is a lightweight, simple few-shot algorithm that can be applied for a wide range of applications, such as healthcare, medical, and security surveillance. The official implementation code of our Few-shot Cosine Transformer is available at https://github.com/vinuni-vishc/Few-Shot-Cosine-Transforme