Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds

Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.Comment: 49 page

A user's guide to optimal transport

This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal transportation, the first and second order differential calculus in the Wasserstein space and the synthetic theory of metric measure spaces with Ricci curvature bounded from below

The sharp-interface limit for the Navier--Stokes--Korteweg equations

We investigate the sharp-interface limit for the Navier--Stokes--Korteweg model, which is an extension of the compressible Navier--Stokes equations. By means of compactness arguments, we show that solutions of the Navier--Stokes--Korteweg equations converge to solutions of a physically meaningful free-boundary problem. Assuming that an associated energy functional converges in a suitable sense, we obtain the sharp-interface limit at the level of weak solutions

Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below

We prove existence and uniqueness of optimal maps on $RCD^*(K,N)$ spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation. \ua9 2015, Mathematica Josephina, Inc

Congested traffic equilibria and degenerate anisotropic PDEs

Congested traffic problems on very dense networks lead, at the limit, to minimization problems posed on measures on curves as shown in Baillon and Carlier (Netw. Heterogenous Media 7: 219--241, 2012). Here, we go one step further by showing that these problems can be reformulated in terms of the minimization of an integral functional over a set of vector fields with prescribed divergence. We prove a Sobolev regularity result for their minimizers despite the fact that the Euler-Lagrange equation of the dual is highly degenerate and anisotropic. This somehow extends the analysis of Brasco et al. (J. Math. Pures Appl. 93: 652--671, 2010) to the anisotropic case

A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes

A Lagrangian numerical scheme for solving nonlinear degenerate Fokker{Planck equations in space dimensions d>2 is presented. It applies to a large class of nonlinear diffusion equations, whose dynamics are driven by internal energies and given external potentials, e.g. the porous medium equation and the fast diffusion equation. The key ingredient in our approach is the gradient ow structure of the dynamics. For discretization of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. Thanks to that time discretisation, the fully discrete solution inherits energy estimates from the original gradient ow, and these lead to weak compactness of the trajectories in the continuous limit. Consistency is analyzed in the planar situation, d = 2. A variety of numerical experiments for the porous medium equation indicates that the scheme is well-adapted to track the growth of the solution's support

SBV regularity of Systems of Conservation Laws and Hamilton-Jacobi Equation

We review the SBV regularity for solutions to hyperbolic systems of conservation laws and Hamilton-Jacobi equations. We give an overview of the techniques involved in the proof, and a collection of related problems concludes the paper

Longitudinal ambulatory measurements of gait abnormality in dystrophin-deficient dogs

Chantier qualité GAInternational audienceABSTRACT: BACKGROUND: This study aimed to measure the gait abnormalities in GRMD (Golden retriever muscular dystrophy) dogs during growth and disease progression using an ambulatory gait analyzer (3D-accelerometers) as a possible tool to assess the effects of a therapeutic intervention. METHODS: Six healthy and twelve GRMD dogs were evaluated twice monthly, from the age of two to nine months. The evolution of each gait variable previously shown to be modified in control and dystrophin-deficient adults was assessed using two-ways variance analysis (age, clinical status) with repeated measurements. A principal component analysis (PCA) was applied to perfect multivariate data interpretation. RESULTS: Speed, stride length, total power and force significantly already decreased (p < 0.01) at the age of 2 months. The other gait variables (stride frequency, relative power distributions along the three axes) became modified at later stages. Using the PCA analysis, a global gait index taking into account the main gait variables was calculated, and was also consistent to detect the early changes in the GRMD gait patterns, as well as the progressive degradation of gait quality. CONCLUSION: The gait variables measured by the accelerometers were sensitive to early detect and follow the gait disorders and mirrored the heterogeneity of clinical presentations, giving sense to monitor gait in GRMD dogs during progression of the disease and pre-clinical therapeutic trials