Search for: All records

In this paper, we study a tumor growth model with nutrients. The model presents dynamic patch solutions due to the incompressibility of the tumor cells. We show that when the nutrients do not diffuse and the cells do not die, the tumor density exhibits regularizing dynamics thanks to an unexpected comparison principle. Using the comparison principle, we provide quantitative $L^1$-contraction estimates and establish the $C^{1,\mathrm{\alpha <\#comment/>}}$-boundary regularity of the tumor patch. Furthermore, whenever the initial nutrient $n_0$either lies entirely above or entirely below the critical value $n_0=1$, we are able to give a complete characterization of the long-time behavior of the system. When $n_0$is constant, we can even describe the dynamics of the full system in terms of some simpler nutrient-free and parameter-free model problems. These results are in sharp contrast to the observed behavior of the models either with nutrient diffusion or with death rate in tumor cells. 

We prove the existence, uniqueness, and regularity of minimizers of a polyconvex functional in two and three dimensions, which corresponds to theH¹-projection of measure-preserving maps. Our result introduces a new criteria on the uniqueness of the minimizer, based on the smallness of the lagrange multiplier. No estimate on the second derivatives of the pressure is needed to get a unique global minimizer. As an application, we construct a minimizing movement scheme to constructL^r-solutions of the Navier–Stokes equation (NSE) for a short time interval. 

We study a family of adversarial multiclass classification problems and provide equivalent reformulations in terms of: 1) a family of generalized barycenter problems introduced in the paper and 2) a family of multimarginal optimal transport problems where the number of marginals is equal to the number of classes in the original classification problem. These new theoretical results reveal a rich geometric structure of adversarial learning problems in multiclass classification and extend recent results restricted to the binary classification setting. A direct computational implication of our results is that by solving either the barycenter problem and its dual, or the MOT problem and its dual, we can recover the optimal robust classification rule and the optimal adversarial strategy for the original adversarial problem. Examples with synthetic and real data illustrate our results. 

Abstract Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoḡlu–Otto and Laux–Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our approach is based on interpreting the Muskat problem as a gradient flow in a product Wasserstein space. This perspective allows us to construct weak solutions via a minimizing movements scheme. Rather than working directly with the singular surface tension force, we instead relax the perimeter functional with the heat content energy approximation of Esedoḡlu–Otto. The heat content energy allows us to show the convergence of the associated minimizing movement scheme in the Wasserstein space, and makes the scheme far more tractable for numerical simulations. Under a typical energy convergence assumption, we show that our scheme converges to weak solutions of the Muskat problem with surface tension. We then conclude the paper with a discussion on some numerical experiments and on equilibrium configurations.