Convergence of an entropic semi-discretization for nonlinear Fokker-Planck equations in Rd

A nonlinear degenerate Fokker-Planck equation in the whole space is analyzed. The existence of solutions to the corresponding implicit Euler scheme is proved, and it is shown that the semi-discrete solution converges to a solution of the continuous problem. Furthermore, the discrete entropy decays monotonically in time and the solution to the continuous problem is unique. The nonlinearity is assumed to be of porous-medium type. For the (given) potential, either a less than quadratic growth condition at infinity is supposed or the initial datum is assumed to be compactly supported. The existence proof is based on regularization and maximum principle arguments. Upper bounds for the tail behavior in space at infinity are also derived in the at-most-quadratic growth case

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

Activation of TRPC1 Channel by Metabotropic Glutamate Receptor mGluR5 Modulates Synaptic Plasticity and Spatial Working Memory

Group I metabotropic glutamate receptors, in particular mGluR5, have been implicated in various forms of synaptic plasticity that are believed to underlie declarative memory. We observed that mGluR5 specifically activated a channel containing TRPC1, an isoform of the canonical family of transient receptor potential (TRPC) channels highly expressed in CA1-3 regions of the hippocampus. TRPC1 is able to form tetrameric complexes with TRPC4 and/or TRPC5 isoforms. TRPC1/4/5 complexes have recently been involved in the efficiency of synaptic transmission in the hippocampus. We therefore used a mouse model devoid of TRPC1 expression to investigate the involvement of mGluR5-TRPC1 pathway in synaptic plasticity and memory formation. Trpc1-/- mice showed alterations in spatial working memory and fear conditioning. Activation of mGluR increased synaptic excitability in neurons from WT but not from Trpc1-/- mice. LTP triggered by a theta burst could not maintain over time in brain slices from Trpc1-/- mice. mGluR-induced LTD was also impaired in these mice. Finally, acute inhibition of TRPC1 by Pico145 on isolated neurons or on brain slices mimicked the genetic depletion of Trpc1 and inhibited mGluR-induced entry of cations and subsequent effects on synaptic plasticity, excluding developmental or compensatory mechanisms in Trpc1-/- mice. In summary, our results indicate that TRPC1 plays a role in synaptic plasticity and spatial working memory processes

Factorization for nonsymmetric operators and exponential H-theorem

Abstract. We present a factorization method for estimating resolvents of nonsymmetric operators in Banach or Hilbert spaces in terms of estimates in another (typically smaller) “reference ” space. This applies to a class of operators writing as a “regularizing ” part (in a broad sense) plus a dissipative part. Then in the Hilbert case we combine this factorization approach with an abstract Plancherel identity on the resolvent into a method for enlarging the functional space of decay estimates on semigroups. In the Banach case, we prove the same result however with some loss on the norm. We then apply these functional analysis approach to several PDEs: the Fokker-Planck and kinetic Fokker-Planck equations, the linear scattering Boltzmann equation in the torus, and, most importantly the linearized Boltzmann equation in the torus (at the price of extra specific work in the latter case). In addition to the abstract method in itself, the main outcome of the paper is indeed the first proof of exponential decay towards global equilibrium (e.g. in terms of the relative entropy) for the full Boltzmann equation for hard spheres, conditionnally to some smoothness and (polynomial) moment estimates. This improves on the result in [12] where the rate was “almost exponential”, that is polynomial with exponent as high as wanted, and solves a long-standing conjecture about the rate of decay in the H-theorem for the nonlinear Boltzmann equation, see for instance [10, 27]

Semidiscretization and long-time asymptotics of nonlinear diffusion equations

We review several results concerning the long-time asymptotics of nonlinear diffusion models based on entropy and mass transport methods. Semidiscretization of these nonlinear diffusion models are proposed and their numerical properties analyzed. We demonstrate the long-time asymptotic results by numerical simulation and we discuss several open problems based on these numerical results. We show that for general nonlinear diffusion equations the long-time asymptotics can be characterized in terms of fixed points of certain maps which are contractions for the euclidean Wasserstein distance. In fact, we propose a new scaling for which we can prove that this family of fixed points converges to the Barenblatt solution for perturbations of homogeneous nonlinearities near zero

Convergence of an entropic semi-discretization for nonlinear Fokker-Planck equations in Rd

A nonlinear degenerate Fokker-Planck equation in the whole space is analyzed. The existence of solutions to the corresponding implicit Euler scheme is proved, and it is shown that the semi-discrete solution converges to a solution of the continuous problem. Furthermore, the discrete entropy decays monotonically in time and the solution to the continuous problem is unique. The nonlinearity is assumed to be of porous-medium type. For the (given) potential, either a less than quadratic growth condition at infinity is supposed or the initial datum is assumed to be compactly supported. The existence proof is based on regularization and maximum principle arguments. Upper bounds for the tail behavior in space at infinity are also derived in the at-most-quadratic growth case

Increasing prevalence of multiple sclerosis in Tuscany, Italy

Background and rationale: An increase of prevalence and incidence of multiple sclerosis (MS) has been reported in several countries, especially taking into account a long-term evaluation. This increasing trend often reflects improved case identification and ascertainment due to the refinement of diagnostic criteria. The aim of this study was to update the prevalence rate of MS in Tuscany (central Italy) as of 2017, and to assess if there has been an increasing trend of prevalence in this Region considering a short period of analysis, from 2014 to 2017. Methods: To capture prevalent cases, a case-finding algorithm based on administrative data, previously created and validated, was used. As data sources, we considered hospital discharge records, drug-dispensing records, disease-specific exemptions from copayment to health care, home and residential long-term care, and inhabitant registry. Results: As of January 1, 2017, 7809 cases were identified, of which 69.4% were females and 30.6% were males. Considering temporal variation, an increasing trend was observed, with standardized rates rising from 189.2 in 2014 to 208.7 per 100,000 in 2017. Conclusions: Results confirm that prevalence increases every year, probably mainly due to the difference between incidence and mortality, resulting in an increasing trend. Moreover, administrative data may accurately identify MS patients in a routinary way and monitor this cohort along disease care pathways