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

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

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

Continuity of Optimal Control Costs and its application to Weak KAM Theory

We prove continuity of certain cost functions arising from optimal control of affine control systems. We give sharp sufficient conditions for this continuity. As an application, we prove a version of weak KAM theorem and consider the Aubry-Mather problems corresponding to these systems.Comment: 23 pages, 1 figures, added explanations in the proofs of the main theorem and the exampl