Analysis of the implicit upwind finite volume scheme with rough coefficients

We study the implicit upwind finite volume scheme for numerically approximating the linear continuity equation in the low regularity DiPerna-Lions setting. That is, we are concerned with advecting velocity fields that are spatially Sobolev regular and data that are merely integrable. We prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique distributional solution of the continuous model is at least 1/2. The numerical error is estimated in terms of logarithmic Kantorovich-Rubinstein distances and provides thus a bound on the rate of weak convergence.Comment: 27 pages. To appear in Numerische Mathemati

Diffusion limited mixing rates in passive scalar advection

We are concerned with flow enhanced mixing of passive scalars in the presence of diffusion. Under the assumption that the velocity gradient is suitably integrable, we provide upper bounds on the exponential rates of mixing and of enhanced dissipation. Our results suggest that there is a crossover from advection dominated to diffusion dominated mixing, and we observe a slow down in the exponential decay rates by (some power of) a logarithm of the diffusivity.Comment: Generalized resul

Eulerian and Lagrangian solutions to the continuity and Euler equations with L1L^1 vorticity

In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of an $L^1$ function, extending the Lagrangian theory in \cite{BouchutCrippa13}. The proof is based on a combination of a stability estimate via optimal transport techniques developed in \cite{Seis16a} and some tools from harmonic analysis introduced in \cite{BouchutCrippa13}. In the second part of the paper, we address a question that arose in \cite{FilhoMazzucatoNussenzveig06}, namely whether 2D Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data has low integrability. We show that this is the case even when the initial vorticity is only in~$L^1$, extending the proof for the $L^p$ case in \cite{CrippaSpirito15}

Nonlocal-interaction equation on graphs: gradient flow structure and continuum limit

We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou-Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of "vertices" is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$^2$IE). We develop the existence theory for the solutions of the NL$^2$IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$^2$IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.Comment: 46 pages. Minor revision with improved presentation and fixed typo