Related papers: Mullins-Sekerka as the Wasserstein flow of the per…
We analyze the convergence rates to a planar interface in the Mullins-Sekerka model by applying a relaxation method based on relationships among distance, energy, and dissipation. The relaxation method was developed by two of the authors in…
The squared Wasserstein distance is a natural quantity to compare probability distributions in a non-parametric setting. This quantity is usually estimated with the plug-in estimator, defined via a discrete optimal transport problem which…
This work presents the discontinuous Galerkin discretization of the consistent splitting scheme proposed by Liu [J. Liu, J. Comp. Phys., 228(19), 2009]. The method enforces the divergence-free constraint implicitly, removing…
We modify the JKO scheme, which is a time discretization of Wasserstein gradient flows, by replacing the Wasserstein distance with more general transport costs on manifolds. We show when the cost function has a mixed Hessian which defines a…
We address the problem of proving a Central Limit Theorem for the empirical optimal transport cost, $\sqrt{n}\{\mathcal{T}_c(P_n,Q)-\mathcal{W}_c(P,Q)\}$, in the semi discrete case, i.e when the distribution $P$ is finitely supported. We…
It is shown that the two-dimensional Mullins-Sekerka problem is well-posed in all subcritical Sobolev spaces $H^r(\mathbb{R})$ with $r\in(3/2,2).$ This is the first result where this issue is established in an unbounded geometry. The…
We present a space-time continuous-Galerkin finite element method for solving incompressible Navier-Stokes equations. To ensure stability of the discrete variational problem, we apply ideas from the variational multi-scale method. The…
We study Fokker--Planck equations with symmetric, positive definite mobility matrices capturing diffusion in heterogeneous environments. A weighted Wasserstein metric is introduced for which these equations are gradient flows. This metric…
Solving Fredholm equations of the first kind is crucial in many areas of the applied sciences. In this work we adopt a probabilistic and variational point of view by considering a minimization problem in the space of probability measures…
We consider a class of time-homogeneous diffusion processes on $\mathbb{R}^{n}$ with common invariant measure but varying volatility matrices. In Euclidean space, we show via stochastic control of the diffusion coefficient that the…
We consider the thresholding scheme, a time discretization for mean curvature flow introduced by Merriman, Bence and Osher. We prove a convergence result in the multi-phase case. The result establishes convergence towards a weak formulation…
In the present paper, we prove that the Wasserstein distance on the space of continuous sample-paths equipped with the supremum norm between the laws of a uniformly elliptic one-dimensional diffusion process and its Euler discretization…
We prove the convergence of a modified Jordan--Kinderlehrer--Otto scheme to a solution to the Fokker--Planck equation in $\Omega \Subset \mathbb R^d$ with general -- strictly positive and temporally constant -- Dirichlet boundary…
In this paper, we address the full discretization of Friedrichs' systems with a two-field structure, such as Maxwell's equations or the acoustic wave equation in div-grad form, cf. [14]. We focus on a discontinuous Galerkin space…
In this paper we consider the flow of two incompressible, viscous and immiscible fluids in a bounded domain, with different densities and viscosities. This model consists of a coupled system of Navier-Stokes and Mullins-Sekerka type parts,…
We present a continuous and a discontinuous linear Finite Element method based on a predictor-corrector scheme for the numerical approximation of the Ericksen-Leslie equations, a model for nematic liquid crystal flow including a non-convex…
We provide the first general result for the asymptotics of the area preserving mean curvature flow in two dimensions showing that flat flow solutions, starting from any bounded set of finite perimeter, converge with exponential rate to a…
We prove the existence of nonnegative weak solutions to a class of second and fourth order nonautonomous nonlinear evolution equations with an explicitly time-dependent mobility function posed on the whole space $\mathbb{R}^d$, for…
We consider the mean-field limit of systems of particles with singular interactions of the type $-\log|x|$ or $|x|^{-s}$, with $0< s<d-2$, and with an additive noise in dimensions $d \geq 3$. We use a modulated-energy approach to prove a…
A new time discretization scheme for the numerical simulation of two-phase flow governed by a thermodynamically consistent diffuse interface model is presented. The scheme is consistent in the sense that it allows for a discrete in time…