Spatial smoothness of the stationary solutions of the 3D Navier--Stokes equations

We consider stationary solutions of the three dimensional Navier--Stokes equations (NS3D) with periodic boundary conditions and driven by an external force which might have a deterministic and a random part. The random part of the force is white in time and very smooth in space. We investigate smoothness properties in space of the stationary solutions. Classical technics for studying smoothness of stochastic PDEs do not seem to apply since global existence of strong solutions is not known. We use the Kolmogorov operator and Galerkin approximations. We first assume that the noise has spatial regularity of order $p$ in the $L^2$ based Sobolev spaces, in other words that its paths are in $H^p$. Then we prove that at each fixed time the law of the stationary solutions is supported by $H^{p+1}$. Then, using a totally different technic, we prove that if the noise has Gevrey regularity then at each fixed time, the law of a stationary solution is supported by a Gevrey space. Some informations on the Kolmogorov dissipation scale are deduced

Markov solutions for the 3D stochastic Navier--Stokes equations with state dependent noise

We construct a Markov family of solutions for the 3D Navier-Stokes equation perturbed by a non degenerate noise. We improve the result of [DPD-NS3D] in two directions. We see that in fact not only a transition semigroup but a Markov family of solutions can be constructed. Moreover, we consider a state dependant noise. Another feature of this work is that we greatly simplify the proofs of [DPD-NS3D]

Exponential mixing for the 3D stochastic Navier--Stokes equations

We study the Navier-Stokes equations in dimension 3 (NS3D) driven by a noise which is white in time. We establish that if the noise is at same time sufficiently smooth and non degenerate in space, then the weak solutions converge exponentially fast to equilibrium. We use a coupling method. The arguments used in dimension two do not apply since, as is well known, uniqueness is an open problem for NS3D. New ideas are introduced. Note however that many simplifications appears since we work with non degenerate noises

Exponential Mixing for Stochastic PDEs: The Non-Additive Case

International audienceWe establish a general criterion which ensures exponential mixing of parabolic Stochastic Partial Differential Equations (SPDE) driven by a non additive noise which is white in time and smooth in space. We apply this criterion on two representative examples: 2D Navier-Stokes (NS) equations and Complex Ginzburg-Landau (CGL) equation with a locally Lipschitz noise. Due to the possible degeneracy of the noise, Doob theorem cannot be applied. Hence a coupling method is used in the spirit of [EMS], [KS3] and [Matt]. Previous results require assumptions on the covariance of the noise which might seem restrictive and artificial. For instance, for NS and CGL, the covariance operator is supposed to be diagonal in the eigenbasis of the Laplacian and not depending on the high modes of the solutions. The method developped in the present paper gets rid of such assumptions and only requires that the range of the covariance operator contains the low modes

Vascular burden as a substrate for higher-level gait disorders in older adults. A review of brain mapping literature

Vascular brain burden, evaluated as white matter hyperintensities (WMH), may explain in part the higher-level gait disorders found in older adults. However, the magnitude and location of WMH as a determinant of higher-level gait disorders remain unknown. The purpose of this review was to determine if the magnitude and distribution of WMH would be associated with the presence of gait disorders in older adults. Medline was searched using the following keywords: "gait", "gait disorders, neurologic", "walking", "cerebrovascular disorders", "leukoaraiosis", "leukoencephalopathies" and "aged". Additional references were reviewed from the bibliographies, and from citation searches on key articles. Observational studies, without language restriction, published between 1995-2011 and exploring simultaneously WMH on MRI and gait performance were selected. Twenty-one studies met the selection criteria. The number of participants per study ranged from 14 to 3301 (35% to 75% female). The total WMH burden was associated with gait disorders in all studies. The largest WMH fractions associated with gait disorders were found in the frontal lobe, the centrum semiovale, the posterior limb of internal capsule, the genu and the splenium of corpus callosum. Gait velocity, stride length and step width were the gait parameters most commonly affected in the presence of WMH. The brain mapping literature supports the hypothesis that a high WMH burden is associated with gait disorders in the course of aging. This could give rise to new strategies for the prevention of higher-level gait disorders and falls in the elderly based on the management of cerebrovascular disease

Ergodic BSDEs and related PDEs with Neumann boundary conditions under weak dissipative assumptions

We study a class of ergodic BSDEs related to PDEs with Neumann boundary conditions. The randomness of the drift is given by a forward process under weakly dissipative assumptions with an invertible and bounded diffusion matrix. Furthermore, this forward process is reflected in a convex subset of $\R^d$ not necessary bounded. We study the link of such EBSDEs with PDEs and we apply our results to an ergodic optimal control problem

Nutrient Biomarker Patterns, Cognitive Function, and Mri Measures of Brain Aging

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