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

Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential

We prove the uniqueness of bounded solutions for the spatially homogeneous Fokker-Planck-Landau equation with a Coulomb potential. Since the local (in time) existence of such solutions has been proved by Arsen'ev-Peskov (1977), we deduce a local well-posedness result. The stability with respect to the initial condition is also checked

On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials

The paper concerns $L^1$- convergence to equilibrium for weak solutions of the spatially homogeneous Boltzmann Equation for soft potentials (-4\le \gm<0), with and without angular cutoff. We prove the time-averaged $L^1$-convergence to equilibrium for all weak solutions whose initial data have finite entropy and finite moments up to order greater than 2+|\gm|. For the usual $L^1$-convergence we prove that the convergence rate can be controlled from below by the initial energy tails, and hence, for initial data with long energy tails, the convergence can be arbitrarily slow. We also show that under the integrable angular cutoff on the collision kernel with -1\le \gm<0, there are algebraic upper and lower bounds on the rate of $L^1$-convergence to equilibrium. Our methods of proof are based on entropy inequalities and moment estimates.Comment: This version contains a strengthened theorem 3, on rate of convergence, considerably relaxing the hypotheses on the initial data, and introducing a new method for avoiding use of poitwise lower bounds in applications of entropy production to convergence problem

Exponential Runge-Kutta methods for stiff kinetic equations

We introduce a class of exponential Runge-Kutta integration methods for kinetic equations. The methods are based on a decomposition of the collision operator into an equilibrium and a non equilibrium part and are exact for relaxation operators of BGK type. For Boltzmann type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes. We give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving. Such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times. The methods also offer favorable properties such as nonnegativity of the solution and entropy inequality. For this reason, as we will show, the methods are suitable both for deterministic as well as probabilistic numerical techniques

Selection of Lactobacillus strains from fermented sausages for their potential use as probiotics.

A rapid screening method was used to isolate potentially probiotic Lactobacillus strains from fermented sausages after enrichment in MRS broth at pH 2.5 followed by bile salt stressing (1% bile salts w/v). One hundred and fifty acid- and bile-resistant strains were selected, avoiding preliminary and time-consuming isolation steps. Strains were further characterized for survival at pH 2.5 for 3 h in phosphate-buffered saline and for growth in the presence of 0.3% bile salts with and without pre-exposure at low pH. Twentyeight strains showed a survival >80% at pH 2.5 for 3 h; moreover, most of the strains were able to grow in the presence of 0.3% bile salts. Low pH and bile resistance was shown to be dependent on both the species, identified by phenotypic and molecular methods, and the strain tested. This is the first report on the direct selection of potentially probiotic lactobacilli from dry fermented sausages. Technologically interesting strains may be used in the future as probiotic starter cultures for novel fermented sausage manufacture