Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections

We consider a stochastic partial differential equation with two logarithmic nonlinearities, with two reflections at 1 and -1 and with a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-time white noise, contains a bi-Laplacian in the drift. The lack of the maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being inspired by the works of Debussche, Gouden\`ege and Zambotti, we obtain existence and uniqueness of solution for initial conditions in the interval $(-1,1)$. Finally, we prove that the unique invariant measure is ergodic, and we give a result of exponential mixing

Non elliptic SPDEs and ambit fields: existence of densities

Relying on the method developed in [debusscheromito2014], we prove the existence of a density for two different examples of random fields indexed by (t,x)\in(0,T]\times \Rd. The first example consists of SPDEs with Lipschitz continuous coefficients driven by a Gaussian noise white in time and with a stationary spatial covariance, in the setting of [dalang1999]. The density exists on the set where the nonlinearity $\sigma$ of the noise does not vanish. This complements the results in [sanzsuess2015] where $\sigma$ is assumed to be bounded away from zero. The second example is an ambit field with a stochastic integral term having as integrator a L\'evy basis of pure-jump, stable-like type.Comment: 23 page

Éducation thérapeutique du patient. Modèles, pratiques et évaluation

International audienceIssu d’une collaboration entre l’Inpes et des acteurs de l’éducation thérapeutique du patient, cet ouvrage rassemble des analyses d’interventions d’éducation thérapeutique mises en place en France et au Québec, dans le cadre de huit maladies chroniques : diabète, obésité, maladies cardio-vasculaires, VIH/sida, asthme, cancer, polyarthrite rhumatoïde et lombalgie. En rendant compte des modèles théoriques qui sous-tendent l’éducation thérapeutique et des démarches mises en œuvre, les contributions mettent au jour une large diversité de pratiques. Qu’il soit professionnel de santé, formateur ou chercheur, le lecteur trouvera ainsi des pistes pour démarrer, développer et évaluer ses actions éducatives. Il trouvera aussi matière à éprouver ses conceptions de la santé et de l’éducation, notamment à travers la découverte de pratiques qui produisent des résultats très encourageants alors qu’elles se réfèrent à des cadres théoriques diversifiés et à des voies différenciées pour penser l’action éducative.Parce qu’elles ne montrent pas l’excellence d’une voie plutôt qu’une autre, ces analyses invitent au développement de nouvelles perspectives d’action et de recherche. L’ouvrage offre ainsi une ouverture précieuse dans un contexte général où l’éducation thérapeutique s’inscrit dans le Code de santé publique, notamment à travers la loi Hôpital, patients, santé et territoires du 21 juillet 2009, qui en reconnaît l’importance pour l’amélioration de l’état de santé des personnes, en particulier de celles atteintes d’une maladie chronique

Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term

The paper is devoted to a modification of the classical Cahn-Hilliard equation proposed by some physicists. This modification is obtained by adding the second time derivative of the order parameter multiplied by an inertial coefficient which is usually small in comparison to the other physical constants. The main feature of this equation is the fact that even a globally bounded nonlinearity is "supercritical" in the case of two and three space dimensions. Thus the standard methods used for studying semilinear hyperbolic equations are not very effective in the present case. Nevertheless, we have recently proven the global existence and dissipativity of strong solutions in the 2D case (with a cubic controlled growth nonlinearity) and for the 3D case with small inertial coefficient and arbitrary growth rate of the nonlinearity. The present contribution studies the long-time behavior of rather weak (energy) solutions of that equation and it is a natural complement of the results of our previous papers. Namely, we prove here that the attractors for energy and strong solutions coincide for both the cases mentioned above. Thus, the energy solutions are asymptotically smooth. In addition, we show that the non-smooth part of any energy solution decays exponentially in time and deduce that the (smooth) exponential attractor for the strong solutions constructed previously is simultaneously the exponential attractor for the energy solutions as well

Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes

We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It is found that the rate of weak convergence is twice that of strong convergence. Furthermore, in contrast to the parabolic case, higher order schemes in time, such as the Crank-Nicolson scheme, are worthwhile to use if the solution is not very regular. Finally we apply the theory to parabolic equations and detail a weak error estimate for the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic heat equation

Stochastic attractors for shell phenomenological models of turbulence

Recently, it has been proposed that the Navier-Stokes equations and a relevant linear advection model have the same long-time statistical properties, in particular, they have the same scaling exponents of their structure functions. This assertion has been investigate rigorously in the context of certain nonlinear deterministic phenomenological shell model, the Sabra shell model, of turbulence and its corresponding linear advection counterpart model. This relationship has been established through a "homotopy-like" coefficient $\lambda$ which bridges continuously between the two systems. That is, for $\lambda=1$ one obtains the full nonlinear model, and the corresponding linear advection model is achieved for $\lambda=0$. In this paper, we investigate the validity of this assertion for certain stochastic phenomenological shell models of turbulence driven by an additive noise. We prove the continuous dependence of the solutions with respect to the parameter $\lambda$. Moreover, we show the existence of a finite-dimensional random attractor for each value of $\lambda$ and establish the upper semicontinuity property of this random attractors, with respect to the parameter $\lambda$. This property is proved by a pathwise argument. Our study aims toward the development of basic results and techniques that may contribute to the understanding of the relation between the long-time statistical properties of the nonlinear and linear models

Stochastic Reaction-diffusion Equations Driven by Jump Processes

We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative functional. The drift in the equations contains a dissipative nonlinearity of polynomial growth.Comment: See journal reference for teh final published versio

Analysis of equilibrium states of Markov solutions to the 3D Navier-Stokes equations driven by additive noise

We prove that every Markov solution to the three dimensional Navier-Stokes equation with periodic boundary conditions driven by additive Gaussian noise is uniquely ergodic. The convergence to the (unique) invariant measure is exponentially fast. Moreover, we give a well-posedness criterion for the equations in terms of invariant measures. We also analyse the energy balance and identify the term which ensures equality in the balance.Comment: 32 page

Global Existence and Regularity for the 3D Stochastic Primitive Equations of the Ocean and Atmosphere with Multiplicative White Noise

The Primitive Equations are a basic model in the study of large scale Oceanic and Atmospheric dynamics. These systems form the analytical core of the most advanced General Circulation Models. For this reason and due to their challenging nonlinear and anisotropic structure the Primitive Equations have recently received considerable attention from the mathematical community. In view of the complex multi-scale nature of the earth's climate system, many uncertainties appear that should be accounted for in the basic dynamical models of atmospheric and oceanic processes. In the climate community stochastic methods have come into extensive use in this connection. For this reason there has appeared a need to further develop the foundations of nonlinear stochastic partial differential equations in connection with the Primitive Equations and more generally. In this work we study a stochastic version of the Primitive Equations. We establish the global existence of strong, pathwise solutions for these equations in dimension 3 for the case of a nonlinear multiplicative noise. The proof makes use of anisotropic estimates, $L^{p}_{t}L^{q}_{x}$ estimates on the pressure and stopping time arguments.Comment: To appear in Nonlinearit