We study the McKean-Vlasov equationwith periodic boundary conditions on the torus. We first study the global asymptotic stability of the homogeneous steady state. We then focus our attention on the stationary system, and prove the existence of nontrivial solutions branching from the homogeneous steady state, through possibly infinitely many bifurcations, under appropriate assumptions on the interaction potential. We also provide sufficient conditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase these results by applying them to several examples of interaction potentials such as the noisy Kuramoto model for synchronisation, the Keller-Segel model for bacterial chemotaxis, and the noisy Hegselmann-Krausse model for opinion dynamics. 13 3.1. Trend to equilibrium in relative entropy 13 3.2. Linear stability analysis 14 4. Bifurcation theory 15 5. Phase transitions for the McKean-Vlasov equation 20 5.1. Discontinuous transition points 22 5.2. Continuous transition points 24 6. Applications 31 6.1. The generalised Kuramoto model 31 6.2. The noisy Hegselmann-Krause model for opinion dynamics 34 6.3. The Onsager model for liquid crystals 34 6.4. The Barré-Degond-Zatorska model for interacting dynamical networks 36 6.5. The Keller-Segel model for bacterial chemotaxis 36 Appendix A. Results from bifurcation theory 39 References 41

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$$^2$$ 2 IE). We develop the existence theory for the solutions of the NL$$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$$^2$$ 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.

We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian H : R n → R in the regime of low temperature ε. We proof the Eyring-Kramers formula for the optimal constant in the Poincaré (PI) and logarithmic Sobolev inequality (LSI) for the associated generator L = ε∆ − ∇H · ∇ of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald et al. [Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 302-351] and of the mean-difference estimate introduced by Chafaï and Malrieu [Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 72-96]. The Eyring-Kramers formula follows as a simple corollary from two main ingredients: The first one shows that the PI and LSI constant of the diffusion restricted to metastable regions corresponding to the local minima scales well in ε. This mimics the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of a mean-difference by a weighted transport distance. It contains the main contribution to the PI and LSI constant, resulting from exponentially long waiting times of jumps between metastable states of the diffusion.

Abstract. This paper is concerned with the numerical analysis of the explicit upwind finite volume scheme for numerically solving continuity equations. We are interested in the case where the advecting velocity field has spatial Sobolev regularity and initial data are merely integrable. We estimate the error between approximate solutions constructed by the upwind scheme and distributional solutions of the continuous problem in a Kantorovich-Rubinstein distance, which was recently used for stability estimates for the continuity equation [23]. Restricted to Cartesian meshes, our estimate shows that the rate of weak convergence is at least of order 1/2 in the mesh size. The proof relies on a probabilistic interpretation of the upwind scheme [9]. We complement the weak convergence result with an example that illustrates that for rough initial data no rates can be expected in strong norms. The same example suggests that the weak order 1/2 rate is optimal.Key words. continuity equation, finite volume scheme, Kantorovich-Rubinstein, order of convergence, stability, upwind AMS subject classifications. 65M08, 65M15, 65M751. Introduction. This paper is concerned with the numerical analysis of the explicit upwind finite volume scheme for solving linear conservative transport equations. We are interested in situations in which the coefficients in the equation are rough, but still within the range in which the associated Cauchy problem is well-posed. To be more specific, we consider nearly incompressible advecting velocity fields with spatial Sobolev regularity and configurations that are integrable but not necessarily bounded. This is the setting studied by DiPerna and Lions in their original paper [12].The goal of this work is an estimate of the error of the numerical scheme. In our main result, we show that the rate of convergence of the approximate solution given by the explicit upwind scheme towards the unique weak solution of the continuous problem is at least of order 1/2 in the mesh size, uniformly in time. Our bound is valid for uniform Cartesian meshes 1 only, but possible extensions to more general meshes are discussed. To measure the numerical error we use nonstandard distances from the theory of optimal mass transportation which appear to be natural in the context of continuity equations [22,23,24]. As these distances metrize weak convergence, the present paper provides a bound on the order of weak convergence. We will moreover see that, in general, strong convergence rates cannot be expected for rough initial data. In this sense, the choice of weak convergence measures is optimal. Our computations moreover suggest that the order 1/2 rate is sharp.Considering coefficients under low regularity assumptions appears to be natural in the context of fluid dynamics, for instance, for problems described by compressible or incompressible inhomogeneous Navier-Stokes equations, or engineering questions related to fluid mixing, which attracted much interest recently [26]. The present work can be considered as a first step t...