Lyapunov functionals for boundary-driven nonlinear drift-diffusions

We exhibit a large class of Lyapunov functionals for nonlinear drift-diffusion equations with non-homogeneous Dirichlet boundary conditions. These are generalizations of large deviation functionals for underlying stochastic many-particle systems, the zero range process and the Ginzburg-Landau dynamics, which we describe briefly. As an application, we prove linear inequalities between such an entropy-like functional and its entropy production functional for the boundary-driven porous medium equation in a bounded domain with positive Dirichlet conditions: this implies exponential rates of relaxation related to the first Dirichlet eigenvalue of the domain. We also derive Lyapunov functions for systems of nonlinear diffusion equations, and for nonlinear Markov processes with non-reversible stationary measures

Association between spondylolisthesis and L5 fracture in patients with osteogenesis imperfecta

To investigate if an association between spondylolisthesis and L5 fracture occurs in patients affected by Osteogenesis Imperfecta (O.I.). Methods Anteroposterior and lateral radiograms were performed on the sample (38 O.I. patients, of whom 19 presenting listhesis); on imaging studies spondylolisthesis was quantified according to the Meyerding classification. Genant’s semiquantitative classification was applied on lateral view to evaluate the L5 fractures; skeleton spinal morphometry (MXA) was carried out on the same images to collect quantitative data comparable and superimposable to Genant’s classification. The gathered information were analyzed through statistical tests (O.R., χ 2 test, Fisher’s test, Pearson’s correlation coefficient). Results The prevalence of L5 fractures is 73.7 % in O.I. patients with spondylolisthesis and their risk of experiencing such a fracture is twice than O.I. patients without listhesis (OR 2.04). Pearson’s χ 2 test demonstrates an association between L5 spondylolisthesis and L5 fracture, especially with moderate, posterior fractures (p = 0.017) and primarily in patients affected by type IV O.I. Conclusions Spondylolisthesis represents a risk factor for the development of more severe and biconcave/posterior type fractures of L5 in patients suffering from O.I., especially in type IV. This fits the hypothesis that the anterior sliding of the soma of L5 alters the dynamics of action of the load forces, localizing them on the central and posterior heights that become the focus of the stress due to movement of flexion–extension and twisting of the spine. As a result, there is greater probability of developing an important subsidence of the central and posterior walls of the soma

On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity

We prove an inequality on the Wasserstein distance with quadratic cost between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, from which we deduce some uniqueness results. In particular, we obtain a local (in time) well-posedness result in the case of (possibly very) soft potentials. A global well-posedeness result is shown for all regularized hard and soft potentials without angular cutoff. Our uniqueness result seems to be the first one applying to a strong angular singularity, except in the special case of Maxwell molecules. Our proof relies on the ideas of Tanaka: we give a probabilistic interpretation of the Boltzmann equation in terms of a stochastic process. Then we show how to couple two such processes started with two different initial conditions, in such a way that they almost surely remain close to each other

Evaluation of the urban tile in MOSES using surface energy balance observations

The UK Met Office has introduced a new scheme for its urban tile in MOSES 2.2 (Met Office Surface Exchange Scheme version 2.2), which is currently implemented within the operational Met Office weather forecasting model. Here, the performance of the urban tile is evaluated in two urban areas: the historic core of downtown Mexico City and a light industrial site in Vancouver, Canada. The sites differ in terms of building structures and mean building heights. In both cases vegetation cover is less than 5%. The evaluation is based on surface energy balance flux measurements conducted at approximately the blend- ing height, which is the location where the surface scheme passes flux data into the atmo- spheric model. At both sites, MOSES 2.2 correctly simulates the net radiation, but there are discrepancies in the partitioning of turbulent and storage heat fluxes between predicted and observed values. Of the turbulent fluxes, latent heat fluxes were underpredicted by about one order of magnitude. Multiple model runs revealed MOSES 2.2 to be sensitive to changes in the canopy heat storage and in the ratio between the aerodynamic roughness length and that for heat transfer (temperature). Model performance was optimum with heat capacity values smaller than those generally considered for these sites. The results suggest that the current scheme is probably too simple, and that improvements may be obtained by increasing the complexity of the model

Nonlocal Aggregation Models: A Primer of Swarm Equilibria

Biological aggregations such as fish schools, bird flocks, bacterial colonies, and insect swarms have characteristic morphologies governed by the group members\u27 intrinsic social interactions with each other and by their interactions with the external environment. Starting from a simple discrete model treating individual organisms as point particles, we derive a nonlocal partial differential equation describing the evolving population density of a continuum aggregation. To study equilibria and their stability, we use tools from the calculus of variations. In one spatial dimension, and for several choices of social forces, external forces, and domains, we find exact analytical expressions for the equilibria. These solutions agree closely with numerical simulations of the underlying discrete model. The analytical solutions provide a sampling of the wide variety of equilibrium configurations possible within our general swarm modeling framework, and include features such as spatial localization with compact support, mass concentrations, and discontinuous density jumps at the edge of the group. We apply our methods to a model of locust swarms, which in nature are observed to consist of a concentrated population on the ground separated from an airborne group. Our model can reproduce this configuration; in this case quasi-two-dimensionality of the locust swarm plays a critical role

Behavioral Interactions Between Japanese Beetle (Coleoptera: Scarabaeidae) Grubs and an Entomopathogenic Nematode (Nematoda: Heterorhabditidae) within Turf Microcosms

Distribution of Japanese beetle, Popillia japonica Newman, grubs and dispersal of an entomopathogenic nematode, Heterorhabditis bacteriophora Poinar ‘Oswego' strain (an isolate from New York state), were examined for 5 wk within soil-filled flats containing grass. Japanese beetle grubs uniformly dispersed to all sections of the flats not infested with H. bacteriophora ‘Oswego' strain. In flats infested with H. bacteriophora ‘Oswego' strain, however, greater proportions of Japanese beetle grubs were recovered in sections near the nematode release site or center sections of the flats. H. bacteriophora ‘Oswego' strain dispersed to all sections of the flats but dispersed more rapidly within the flats infested with Japanese beetle grubs than in flats not infested with Japanese beetle grub

Effective dynamics using conditional expectations

The question of coarse-graining is ubiquitous in molecular dynamics. In this article, we are interested in deriving effective properties for the dynamics of a coarse-grained variable $\xi(x)$, where $x$ describes the configuration of the system in a high-dimensional space $\R^n$, and $\xi$ is a smooth function with value in $\R$ (typically a reaction coordinate). It is well known that, given a Boltzmann-Gibbs distribution on $x \in \R^n$, the equilibrium properties on $\xi(x)$ are completely determined by the free energy. On the other hand, the question of the effective dynamics on $\xi(x)$ is much more difficult to address. Starting from an overdamped Langevin equation on $x \in \R^n$, we propose an effective dynamics for $\xi(x) \in \R$ using conditional expectations. Using entropy methods, we give sufficient conditions for the time marginals of the effective dynamics to be close to the original ones. We check numerically on some toy examples that these sufficient conditions yield an effective dynamics which accurately reproduces the residence times in the potential energy wells. We also discuss the accuracy of the effective dynamics in a pathwise sense, and the relevance of the free energy to build a coarse-grained dynamics

Evolution models for mass transportation problems

We present a survey on several mass transportation problems, in which a given mass dynamically moves from an initial configuration to a final one. The approach we consider is the one introduced by Benamou and Brenier in [5], where a suitable cost functional $F(\rho,v)$, depending on the density $\rho$ and on the velocity $v$ (which fulfill the continuity equation), has to be minimized. Acting on the functional $F$ various forms of mass transportation problems can be modeled, as for instance those presenting congestion effects, occurring in traffic simulations and in crowd motions, or concentration effects, which give rise to branched structures.Comment: 16 pages, 14 figures; Milan J. Math., (2012

Celebrating Cercignani's conjecture for the Boltzmann equation

Cercignani's conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann's nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s.Comment: This paper is dedicated to the memory of the late Carlo Cercignani, powerful mind and great scientist, one of the founders of the modern theory of the Boltzmann equation. 24 pages. V2: correction of some typos and one ref. adde