Relaxation of the Cahn-Hilliard equation with singular single-well potential and degenerate mobility

The degenerate Cahn-Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short range attraction and long range repulsion effects. In this framework, we consider the usual Cahn-Hilliard equation with a degenerate double-well potential and degenerate mobility. These degeneracies induce numerous difficulties, in particuler for its numerical simulation. To overcome these issues, we propose a relaxation system formed of two second order equations which can be solved with standard packages. This system is endowed with an energy and an entropy structure compatible with the limiting equation. Here, we study the theoretical properties of this system; global existence and convergence of the relaxed system to the degenerate Cahn-Hilliard equation. We also study the long time asymptotics which interest relies on the numerous possible steady states with given mass

A nonnegativity preserving scheme for the relaxed Cahn-Hilliard equation with single-well potential and degenerate mobility

We propose and analyze a finite element approximation of the relaxed Cahn-Hilliard equation with singular single-well potential of Lennard-Jones type and degenerate mobility that is energy stable and nonnegativity preserving. The Cahn-Hilliard model has recently been applied to model evolution and growth for living tissues: although the choices of degenerate mobility and singular potential are biologically relevant, they induce difficulties regarding the design of a numerical scheme. We propose a finite element scheme and we show that it preserves the physical bounds of the solutions thanks to an upwind approach adapted to the finite element method. Moreover, we show well-posedness, energy stability properties, and convergence of solutions of the numerical scheme. Finally, we validate our scheme by presenting numerical simulations in one and two dimensions

A Positivity-Preserving Finite Element Scheme for the Relaxed Cahn-Hilliard Equation with Single-Well Potential and Degenerate Mobility

We propose and analyse a finite element approximation of the Cahn-Hilliard equation regularised in space with single-well potential of Lennard-Jones type and degenerate mobility. The Cahn-Hilliard model has recently been applied to model evolution and growth for living tissues: although the choices of degenerate mobility and singular potential are biologically relevant, they induce difficulties regarding the design of a numerical scheme. We propose a finite element scheme in one and two dimensions and we show that it preserves the physical bounds of the solutions thanks to an upwind approach adapted to the finite elements method. Moreover, we show well-posedness, energy stability properties and convergence of solutions to the numerical scheme. Finally, numerical simulations in one and two dimensions are presented

Scalar auxiliary variable finite element scheme for the parabolic-parabolic Keller-Segel model

We describe and analyze a finite element numerical scheme for the parabolic-parabolic Keller-Segel model. The scalar auxiliary variable method is used to retrieve the monotonic decay of the energy associated with the system at the discrete level. This method relies on the interpretation of the Keller-Segel model as a gradient flow. The resulting numerical scheme is efficient and easy to implement. We show the existence of a unique non-negative solution and that a modified discrete energy is obtained due to the use of the SAV method. We also prove the convergence of the discrete solutions to the ones of the weak form of the continuous Keller-Segel model

Transformations rédox et spéciation du Hg dans la neige et les eaux de surface de l'extrême arctique et de régions tempérées

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal

Cryptomelane formation from nanocrystalline vernadite precursor: a high energy X-ray scattering and transmission electron microscopy perspective on reaction mechanisms

International audienceBackground:Vernadite is a nanocrystalline and turbostratic phyllomanganate which is ubiquitous in the environ‑ment. Its layers are built of (MnO6)8− octahedra connected through their edges and frequently contain vacancies and(or) isomorphic substitutions. Both create a layer charge deficit that can exceed 1 valence unit per layer octahedron and thus induces a strong chemical reactivity. In addition, vernadite has a high affinity for many trace elements (e.g., Co, Ni, and Zn) and possesses a redox potential that allows for the oxidation of redox sensitive elements (e.g., As, Cr, Tl). As a result, vernadite acts as a sink for many trace metal elements. In the environment, vernadite is often found associated with tectomanganates (e.g., todorokite and cryptomelane) of which it is thought to be the precursor. The transformation mechanism is not yet fully understood however and the fate of metals initially contained in vernadite structure during this transformation is still debated. In the present work, the transformation of synthetic vernadite (δ‑MnO2) to synthetic cryptomelane under conditions analogous to those prevailing in soils (dry state, room tempera‑ture and ambient pressure, in the dark) and over a time scale of ~10years was monitored using high‑energy X‑ray scattering (with both Bragg‑rod and pair distribution function formalisms) and transmission electron microscopy

Mining-impacted rice paddies select for Archaeal methylators and reveal a putative (Archaeal) regulator of mercury methylation

ABSTRACT: Methylmercury (MeHg) is a microbially produced neurotoxin derived from inorganic mercury (Hg), which accumulation in rice represents a major health concern to humans. However, the microbial control of MeHg dynamics in the environment remains elusive. Here, leveraging three rice paddy fields with distinct concentrations of Hg (Total Hg (THg): 0.21−513 mg kg−1 dry wt. soil; MeHg: 1.21−6.82 ng g−1 dry wt. soil), we resorted to metagenomics to determine the microbial determinants involved in MeHg production under contrasted contamination settings. We show that Hg methylating Archaea, along with methane-cycling genes, were enriched in severely contaminated paddy soils. Metagenome-resolved Genomes of novel putative Hg methylators belonging to Nitrospinota (UBA7883), with poorly resolved taxonomy despite high completeness, showed evidence of facultative anaerobic metabolism and adaptations to fluctuating redox potential. Furthermore, we found evidence of environmental filtering effects that influenced the phylogenies of not only hgcA genes under different THg concentrations, but also of two housekeeping genes, rpoB and glnA, highlighting the need for further experimental validation of whether THg drives the evolution of hgcAB. Finally, assessment of the genomic environment surrounding hgcAB suggests that this gene pair may be regulated by an archaeal toxin-antitoxin (TA) system, instead of the more frequently found arsR-like genes in bacterial methylators. This suggests the presence of distinct hgcAB regulation systems in bacteria and archaea. Our results support the emerging role of Archaea in MeHg cycling under mining-impacted environments and shed light on the differential control of the expression of genes involved in MeHg formation between Archaea and Bacteria

Mercury in the marine environment of the Canadian Arctic: Review of recent findings

AbstractThis review summarizes data and information which have been generated on mercury (Hg) in the marine environment of the Canadian Arctic since the previous Canadian Arctic Contaminants Assessment Report (CACAR) was released in 2003. Much new information has been collected on Hg concentrations in marine water, snow and ice in the Canadian Arctic. The first measurements of methylation rates in Arctic seawater indicate that the water column is an important site for Hg methylation. Arctic marine waters were also found to be a substantial source of gaseous Hg to the atmosphere during the ice-free season. High Hg concentrations have been found in marine snow as a result of deposition following atmospheric mercury depletion events, although much of this Hg is photoreduced and re-emitted back to the atmosphere. The most extensive sampling of marine sediments in the Canadian Arctic was carried out in Hudson Bay where sediment total Hg (THg) concentrations were low compared with other marine regions in the circumpolar Arctic. Mass balance models have been developed to provide quantitative estimates of THg fluxes into and out of the Arctic Ocean and Hudson Bay.Several recent studies on Hg biomagnification have improved our understanding of trophic transfer of Hg through marine food webs. Over the past several decades, Hg concentrations have increased in some marine biota, while other populations showed no temporal change. Marine biota also exhibited considerable geographic variation in Hg concentrations with ringed seals, beluga and polar bears from the Beaufort Sea region having higher Hg concentrations compared with other parts of the Canadian Arctic. The drivers of these variable patterns of Hg bioaccumulation, both regionally and temporally, within the Canadian Arctic remain unclear. Further research is needed to identify the underlying processes including the interplay between biogeochemical and food web processes and climate change