Weak convergence results for inhomogeneous rotating fluid equations

We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vector B(x): this is a generalization of the usual rotating fluid model (where B is constant). We prove the weak convergence of Leray--type solutions towards a vector field which satisfies the usual 2D Navier--Stokes equation in the regions of space where B is constant, with Dirichlet boundary conditions, and a heat--type equation elsewhere. The method of proof uses weak compactness arguments

Mathematical study of the betaplane model: Equatorial waves and convergence results

We are interested in a model of rotating fluids, describing the motion of the ocean in the equatorial zone. This model is known as the Saint-Venant, or shallow-water type system, to which a rotation term is added whose amplitude is linear with respect to the latitude; in particular it vanishes at the equator. After a physical introduction to the model, we describe the various waves involved and study in detail the resonances associated with those waves. We then exhibit the formal limit system (as the rotation becomes large), obtained as usual by filtering out the waves, and prove its wellposedness. Finally we prove three types of convergence results: a weak convergence result towards a linear, geostrophic equation, a strong convergence result of the filtered solutions towards the unique strong solution to the limit system, and finally a "hybrid" strong convergence result of the filtered solutions towards a weak solution to the limit system. In particular we obtain that there are no confined equatorial waves in the mean motion as the rotation becomes large.Comment: Revised version after referee's comments. Accepted for publication in M\'{e}moires de la Soci\'{e}t\'{e} Math\'{e}matique de Franc

Semiclassical and spectral analysis of oceanic waves

In this work we prove that the shallow water flow, subject to strong wind forcing and linearized around an adequate stationary profile, develops for large times closed trajectories due to the propagation of Rossby waves, while Poincar\'e waves are shown to disperse. The methods used in this paper involve semi-classical analysis and dynamical systems for the study of Rossby waves, while some refined spectral analysis is required for the study of Poincar\'e waves, due to the large time scale involved which is of diffractive type

Proderm technology: a water- based lipid delivery system for dermatitis that penetrates viable epidermis and has antibacterial effects.

BackgroundA defective skin barrier and bacterial colonization are two important factors in maintenance and progression of atopic dermatitis and chronic allergic/irritant hand dermatitis. A water-based lipid delivery system containing physiologic lipids was previously shown to be a useful adjunct in the treatment of hand dermatitis. We tested the ability of this formulation to penetrate into the viable epidermis and in addition assessed its antibacterial properties.MethodsEpidermal penetration of the product was assessed by fluorescence microscopy. Recovery of Escherichia coli and Staphylococcus aureus MRSA from skin treated with Neosalus® foam was quantified.ResultsComponents of Neosalus® penetrated the stratum corneum and were distributed throughout the viable epidermis. Neosalus® significantly decreased recovery of both Staphylococcus aureus and Escherichia coli from the skin surface.ConclusionsThe ability of components of Neosalus® to be taken up into the viable epidermis and potentially made available for incorporation into the barrier lipids, combined with antibacterial properties, indicate that this formulation may be valuable not only in chronic hand dermatitis, but also in various other forms of dermatitis.Trial registrationCurrent Controlled Trials ISRCTN18191379 , 28/12/2018, retrospectively registered

The Brownian motion as the limit of a deterministic system of hard-spheres

We provide a rigorous derivation of the brownian motion as the limit of a deterministic system of hard-spheres as the number of particles $N$ goes to infinity and their diameter $\varepsilon$ simultaneously goes to $0$, in the fast relaxation limit $\alpha = N\varepsilon^{d-1}\to \infty$ (with a suitable diffusive scaling of the observation time). As suggested by Hilbert in his sixth problem, we rely on a kinetic formulation as an intermediate level of description between the microscopic and the fluid descriptions: we use indeed the linear Boltzmann equation to describe one tagged particle in a gas close to global equilibrium. Our proof is based on the fundamental ideas of Lanford. The main novelty here is the detailed study of the branching process, leading to explicit estimates on pathological collision trees

On the propagation of oceanic waves driven by a strong macroscopic flow

In this work we study oceanic waves in a shallow water flow subject to strong wind forcing and rotation, and linearized around a inhomogeneous (non zonal) stationary profile. This extends the study~\cite{CGPS}, where the profile was assumed to be zonal only and where explicit calculations were made possible due to the 1D setting. Here the diagonalization of the system, which allows to identify Rossby and Poincaré waves, is proved by an abstract semi-classical approach. The dispersion of Poincaré waves is also obtained by a more abstract and more robust method using Mourre estimates. Only some partial results however are obtained concerning the Rossby propagation, as the two dimensional setting complicates very much the study of the dynamical system

From Hard Sphere Dynamics to the Stokes–Fourier Equations: An Analysis of the Boltzmann–Grad Limit

to appear, Annals of PDEsInternational audienceWe derive the linear acoustic and Stokes-Fourier equations as the limiting dynamics of a system of N hard spheres of diameter $\epsilon$ in two space dimensions, when N $\rightarrow$ $\infty$, $\epsilon$ $\rightarrow$ 0, N $\epsilon$ = $\alpha$ $\rightarrow$ $\infty$, using the linearized Boltzmann equation as an intermediate step. Our proof is based on Lanford's strategy [18], and on the pruning procedure developed in [5] to improve the convergence time to all kinetic times with a quantitative control which allows us to reach also hydrodynamic time scales. The main novelty here is that uniform L 2 a pri-ori estimates combined with a subtle symmetry argument provide a weak version of chaos, in the form of a cumulant expansion describing the asymptotic decorrelation between the particles. A refined geometric analysis of recollisions is also required in order to discard the possibility of multiple recollisions

Development of a pig jejunal explant culture for studying the gastrointestinal toxicity of the mycotoxin deoxynivalenol: histopathological analysis

The digestive tract is a target for the mycotoxin deoxynivalenol (DON), a major cereals grain contaminant of public health concern in Europe and North America. Pig, the most sensitive species to DON toxicity, can be regarded as the most relevant animal model for studying the intestinal effects of DON. A pig jejunal explants culture was developed to assess short-term effects of DON. In a first step, jejunal explants from 9-13 week-old and from 4-5 week-old pigs were cultured in vitro for up to 8 hours. Explants from younger animals were better preserved after 8 hours, as assessed by morphological scores and by villi lengths. In a second step, dose-related alterations of the jejunal tissue were observed, including shortened and coalescent villi, lysis of enterocytes, oedema. After 4h of DON exposure of explants from 4-5 week-old pigs, a no-effect concentration level of 1 µM was estimated (corresponding to diet contaminated with 0.3 mg DON/kg) based on morphological scores, and of 0.2 µM based on villi lengths. In conclusion, our data indicate that pig intestinal explants represent a relevant and sensitive model to investigate the effects of food contaminants