A probabilistic approach to large time behaviour of viscosity solutions of parabolic equations with Neumann boundary conditions

This paper is devoted to the study of the large time behaviour of viscosity solutions of parabolic equations with Neumann boundary conditions. This work is the sequel of [13] in which a probabilistic method was developped to show that the solution of a parabolic semilinear PDE behaves like a linear term $\lambda T$ shifted with a function $v$, where $(v,\lambda)$ is the solution of the ergodic PDE associated to the parabolic PDE. We adapt this method in finite dimension by a penalization method in order to be able to apply an important basic coupling estimate result and with the help of a regularization procedure in order to avoid the lack of regularity of the coefficients in finite dimension. The advantage of our method is that it gives an explicit rate of convergence

Sources, fate, and pathways of Leeuwin Current water in the Indian Ocean and Great Australian Bight: A Lagrangian study in an eddy-resolving ocean model

The Leeuwin Current is the dominant circulation feature in the eastern Indian Ocean, transporting tropical and subtropical water southward. While it is known that the Leeuwin Current draws its water from a multitude of sources, existing Indian Ocean circulation schematics have never quantified the fluxes of tropical and subtropical source water flowing into the Leeuwin Current. This paper uses virtual Lagrangian particles to quantify the transport of these sources along the Leeuwin Current's mean pathway. Here the pathways and exchange of Leeuwin Current source waters across six coastally bound sectors on the south-west Australian coast are analyzed. This constitutes the first quantitative assessment of Leeuwin Current pathways within an offline, 50 year integration time, eddy-resolving global ocean model simulation. Along the Leeuwin Current's pathway, we find a mean poleward transport of 3.7 Sv in which the tropical sources account for 60-78% of the transport. While the net transport is small, we see large transports flowing in and out of all the offshore boundaries of the Leeuwin Current sectors. Along the Leeuwin Current's pathway, we find that water from the Indonesian Throughflow contributes 50-66% of the seasonal signal. By applying conditions on the routes particles take entering the Leeuwin Current, we find particles are more likely to travel offshore north of 30°S, while south of 30°S, particles are more likely to continue downstream. We find a 0.2 Sv pathway of water from the Leeuwin Current's source regions, flowing through the entire Leeuwin Current pathway into the Great Australian Bight

Bistability induced by generalist natural enemies can reverse pest invasions

Reaction-diffusion analytical modeling of predator-prey systems has shown that specialist natural enemies can slow, stop and even reverse pest invasions, assuming that the prey population displays a strong Allee effect in its growth. Few additional analytical results have been obtained for other spatially distributed predator-prey systems, as traveling waves of non-monotonous systems are notoriously difficult to obtain. Traveling waves have indeed recently been shown to exist in predator-prey systems, but the direction of the wave, an essential item of information in the context of the control of biological invasions, is generally unknown. Preliminary numerical explorations have hinted that control by generalist predators might be possible for prey populations displaying logistic growth. We aimed to formalize the conditions in which spatial biological control can be achieved by generalists, through an analytical approach based on reaction-diffusion equations. The population of the focal prey - the invader - is assumed to grow according to a logistic function. The predator has a type II functional response and is present everywhere in the domain, at its carrying capacity, on alternative hosts. Control, defined as the invader becoming extinct in the domain, may result from spatially independent demographic dynamics or from a spatial extinction wave. Using comparison principles, we obtain sufficient conditions for control and for invasion, based on scalar bistable partial differential equations (PDEs). The searching efficiency and functional response plateau of the predator are identified as the main parameters defining the parameter space for prey extinction and invasion. Numerical explorations are carried out in the region of those control parameters space between the super-and subso-lutions, in which no conclusion about controllability can be drawn on the basis of analytical solutions. The ability of generalist predators to control prey populations with logistic growth lies in the bis-table dynamics of the coupled system, rather than in the bistability of prey-only dynamics as observed for specialist predators attacking prey populations displaying Allee effects. The consideration of space in predator-prey systems involving generalist predators with a parabolic functional response is crucial. Analysis of the ordinary differential equations (ODEs) system identifies parameter regions with monostable (extinction) and bistable (extinction or invasion) dynamics. By contrast, analysis of the associated PDE system distinguishes different and additional regions of invasion and extinction. Depending on the relative positions of these different zones, four patterns of spatial dynamics can be identified : traveling waves of extinction and invasion, pulse waves of extinction and heterogeneous stationary positive solutions of the Turing type. As a consequence, prey control is predicted to be possible when space is considered in additional situations other than those identified without considering space. The reverse situation is also possible. None of these considerations apply to spatial predator-prey systems with specialist natural enemies

Explicit representation and parametrised impacts of under ice shelf seas in the z∗ coordinate ocean model NEMO 3.6

Ice-shelf-ocean interactions are a major source of freshwater on the Antarctic continental shelf and have a strong impact on ocean properties, ocean circulation and sea ice. However, climate models based on the ocean-sea ice model NEMO (Nucleus for European Modelling of the Ocean) currently do not include these interactions in any detail. The capability of explicitly simulating the circulation beneath ice shelves is introduced in the non-linear free surface model NEMO. Its implementation into the NEMO framework and its assessment in an idealised and realistic circum-Antarctic configuration is described in this study. Compared with the current prescription of ice shelf melting (i.e. at the surface), inclusion of open sub-ice-shelf cavities leads to a decrease in sea ice thickness along the coast, a weakening of the ocean stratification on the shelf, a decrease in salinity of high-salinity shelf water on the Ross and Weddell sea shelves and an increase in the strength of the gyres that circulate within the over-deepened basins on the West Antarctic continental shelf. Mimicking the overturning circulation under the ice shelves by introducing a prescribed meltwater flux over the depth range of the ice shelf base, rather than at the surface, is also assessed. It yields similar improvements in the simulated ocean properties and circulation over the Antarctic continental shelf to those from the explicit ice shelf cavity representation. With the ice shelf cavities opened, the widely used "three equation" ice shelf melting formulation, which enables an interactive computation of melting, is tested. Comparison with observational estimates of ice shelf melting indicates realistic results for most ice shelves. However, melting rates for the Amery, Getz and George VI ice shelves are considerably overestimated

Equations différentielles stochastiques rétrogrades ergodiques et applications aux EDP

This thesis deals with the study of ergodic BSDE and their applications to the study of the large time behaviour of solutions to semilinear parabolic PDE. In a first time, we establish some existence and uniqueness results to an ergodic BSDE with Neumann boundary conditions in an unbounded convex set in a weakly dissipative environment. Then we study their link with PDE with Neumann boundary condition and we give an application to an ergodic stochastic control problem. The second part consists of two sections. In the first one, we study the large time bahaviour of mild solutions to semilinear parabolic PDE in infinite dimension by a probabilistic method. This probabilistic method relies on a Basic coupling estimate result which gives us an exponential rate of convergence of the solution toward its asymptote. Let us mention that that this asymptote is fully determined by the solution of the ergodic semilinear PDE associated to the parabolic semilinear PDE. Then, we adapt this method to the sudy of the large time behaviour of viscosity solutions of semilinear parabolic PDE with Neumann boundary condition in a convex and bounded set in finite dimension. By regularization and penalization procedures, we obtain similar results as those obtained in the mild context, especially with an exponential rate of convergence for the solution toward its asymptote.Cette thèse s'intéresse à l'étude des EDSR ergodiques et à leurs applications à l'étude du comportement en temps long des solutions d'EDP paraboliques semi-linéaires. Dans un premier temps, nous établissons des résultats d'existence et d'unicité d'une EDSR ergodique avec conditions de Neumann au bord dans un convexe non borné et dans un environnement faiblement dissipatif. Nous étudions ensuite leur lien avec les EDP avec conditions de Neumann au bord et nous donnons un exemple d'application à un problème de contrôle optimal stochastique. La deuxième partie est constituée de deux sous-parties. Tout d'abord, nous étudions le comportement en temps long des solutions mild d'une EDP parabolique semi-linéaire en dimension infinie par des méthodes probabilistes. Cette méthode probabiliste repose sur une application d'un résultat nommé "Basic coupling estimate" qui nous permet d'obtenir une vitesse de convergence exponentielle de la solution vers sons asymptote. Au passage notons que cette asymptote est entièrement déterminée par la solution de l'EDP ergodique semi-linéaire associée à l'EDP parabolique semi-linéaire initiale. Puis, nous adaptons cette méthode à l'étude du comportement en temps long des solutions de viscosité d'une EDP parabolique semi-linéaire avec condition de Neumann au bord dans un convexe borné en dimension finie. Par des méthodes de régularisation et de pénalisation des coefficients et en utilisant un résultat de stabilité pour les EDSR, nous obtenons des résultats analogues à ceux obtenus dans le contexte mild, avec notamment une vitesse exponentielle de convergence de la solution vers son asymptote

Investissement des ménages : en attendant l’ELAN

Le 4 avril dernier a été présenté en Conseil des ministres le projet de loi « Évolution du logement et aménagement numérique » (ELAN). Depuis l’automne 2017, les thématiques de la construction et du logement sont réapparues dans le débat public lorsque le gouvernement a annoncé sa volonté de faire des économies budgétaires sur les aides à la personne et sur la politique du logement. Alors que l’analyse des déterminants structurels de l’investissement des ménages laissait présager son fort redressement en 2017, il semblerait que les déclarations du gouvernement aient provoqué un certain attentisme sur le marché de la construction. [Premier paragraphe

Le bâtiment a fière allure…

Les chiffres du premier trimestre 2017 publiés le 28 avril par l’INSEE ont confirmé le redressement du secteur de la construction à l’œuvre depuis maintenant plusieurs trimestres : le nombre de logements mis en chantier a atteint 379 000 unités, dépassant ainsi sa moyenne observée depuis 2000 (375 000). Dans le même temps, le nombre de logements autorisés à la construction frôle la barre symbolique des 450 000. [Premier paragraphe

Présidentielle : le logement est-il bien loti ?

La publication du rapport de la Fondation Abbé Pierre en janvier 2017 n’a pas fait exception : le mal-logement continue de progresser en France. Les prix immobiliers sont repartis à la hausse ces derniers trimestres et la publication des premiers résultats de l’Enquête nationale logement de l’INSEE de 2013 font apparaître une dégradation de la situation financière des ménages. Malgré tous ces éléments, la thématique du logement est apparue relativement tardivement dans les débats entourant l’élection présidentielle. Nous tentons ici d’esquisser un panorama des propositions émanant des principaux candidats à l’élection présidentielle sur ce sujet. [Premier paragraphe