Michael Westdickenberg

EULERIAN CALCULUS FOR THE CONTRACTION IN THE WASSERSTEIN DISTANCE

We consider the porous medium equation on a compact Riemannian manifold and give a new proof of the contraction of its semigroup in the Wasserstein distance. This proof is based on the insight that the porous medium equation does not increase the size of infinitesimal perturbations along gradient flow trajectories and on an Eulerian formulation for the Wasserstein distance using smooth curves. Our approach avoids the existence result for optimal transport maps on Riemannian manifolds.

Sticky particle dynamics with interactions

Abstract We consider compressible pressureless fluid flows in Lagrangian coordinates in one space dimension. We assume that the fluid self-interacts through a force field generated by the fluid itself. We explain how this flow can be described by a differential inclusion on the space of transport maps, in particular when a sticky particle dynamics is assumed. We study a discrete particle approximation and we prove global existence and stability results for solutions of this system. In the particular case of the Euler–Poisson system in the attractive regime our approach yields an explicit representation formula for the solutions.

Optimal Transport for the System of Isentropic Euler Equations

We introduce a new variational time discretization for the system of isentropic Euler equations. In each timestep the internal energy is reduced as much as possible, subject to a constraint imposed by a new cost functional that measures the deviation of particles from their characteristic paths.

A New Convergence Proof for Finite Volume Schemes Using the Kinetic Formulation of Conservation Laws

We give a new convergence proof for finite volume schemes approximating scalar conservation laws. The main ingredients of the proof are the kinetic formulation of scalar conservation laws, a discrete entropy inequality, and the velocity averaging technique.

Transport Equations and Multi-D Hyperbolic Conservation Laws

I.- Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields.- II.- A Note on Albertis Rank-One Theorem.- III.- Regularizing Effect of Nonlinearity in Multidimensional Scalar Conservation Laws.

Convergence of Approximate Solutions of Conservation Laws

In this paper we consider convergence of approximate solutions of conservation laws. We start with an overview over the historical developments since the 1950s, and the analytical tools used in this context. Then we present some of our own results on the convergence of numerical approximations, discuss recent related work and open problems.

PROJECTIONS ONTO THE CONE OF OPTIMAL TRANSPORT MAPS AND COMPRESSIBLE FLUID FLOWS

The system of isentropic Euler equations in the potential flow regime can be considered formally as a second order ordinary differential equation on the Wasserstein space of probability measures. This interpretation can be used to derive a variational time discretization. We prove that the approximate solutions generated by this discretization converge to a measure-valued solution of the isentropic Euler equations. The key ingredient is a characterization of the polar cone to the cone of optimal transport maps.

Total oscillation diminishing property for scalar conservation laws

Summary.We prove a BV estimate for scalar conservation laws that generalizes the classical Total Variation Diminishing property. In fact, for any Lipschitz continuous monotone Φ:ℝ→ℝ, we have that |Φ(u)|TV(ℝ) is nonincreasing in time. We call this property Total Oscillation Diminishing because it is in contradiction with the oscillations observed recently in some numerical computations based on TVD schemes. We also show that semi-discrete Total Variation Diminishing finite volume schemes are TOD and that the fully discrete Godunov scheme is TOD.

CONVERGENCE OF THIN FILM APPROXIMATION FOR A SCALAR CONSERVATION LAW

In this paper we consider the thin film approximation of a 1D scalar conservation law with strictly convex flux. We prove that the sequence of approximate solutions converges to the unique Kružkov solution.

The polar cone of the set of monotone maps

We prove that every element of the polar cone to the closed convex cone of monotone transport maps can be represented as the divergence of a measure field taking values in the positive definite matrices.