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{We prove, under certain regularity assumptions on the coefficients, that tangent processes (namely semimartingales d xi(tau) = a dx(tau) + b d tau where a is an antisymmetric matrix) generate flows on the classical Wiener space. Main applications of the result can be found in the study of the geometry of path spaces. (C) 1999 Academic Press.}

{We construct surface type measures on the level sets for the renormalized energy, which is an invariant quantity for the two dimensional periodic Euler flow, and prove the existence of weak solutions living on such level sets. Other classes of invariant measures for the motion are also introduced.}

{A stochastic variational principle for the classical Burgers equation is established. A solution of this equation can be considered as the velocity field of a stochastic process which is a minimum of an energy functional. A family of stochastic constants of the motion, determined in terms of the probability distribution of that process, yields the complete list of symmetries of the Burgers equation.}

{This work consists on the study of flows associated with non-smooth R-d-vector fields, namely concerning existence and uniqueness for almost-every initial condition. It is also proved that the flows avoid some special compact sets. (c) 2005 Elsevier Inc. All rights reserved.}

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{We prove the existence of weak solutions for the forward and backward statistical transport equations associated with the 2D Euler equations. Such solutions can be interpreted, respectively, as a statistical Lagrangian and a statistical Eulerian description of the motion of the fluid.}

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{A stochastic variational principle for the (two dimensional) Navier-Stokes equation is established. The velocity field can be considered as a generalized velocity of a diffusion process with values on the volume preserving diffeomorphism group of the underlying manifold. Navier-Stokes equation is reinterpreted as a perturbed equation of geodesics for the L (2) norm. The method described here should hold as well in higher dimensions.}

{We consider a positive distribution Phi such that Phi defines a probability measure mu = mu Phi on the dual of some real nuclear Frechet space. A large deviation principle is proved for the family \{mu(n), n >= 1\} where mu(n) denotes the image measure of the product measure mu(n)(Phi) under the empirical distribution function L(n). Here the rate function I is defined on the space F(theta)'(N')(+) and agrees with the relative entropy function (H) over tilde (Psi/Phi). As an application, we cite the Gibbs conditioning principle which describes the limiting behaviour as n tends to infinity of the law of k tagged particles Y(1),...,Y(k) under the constraint that L(n)(Y) belongs to some subset A(0).}

{A stochastic representation for the solutions of the Poisson-Vlasov equation is obtained. The representation involves both an exponential and a branching process. The stochastic representation, besides providing an alternative existence proof and an intuitive characterization of the solutions, may also be used to obtain an intrinsic definition of the fluctuations. (c) 2007 Elsevier B.V. All rights reserved.}

Mathematics Subject Classification: 26A33, 76M35, 82B31A stochastic solution is constructed for a fractional generalization of the KPP (Kolmogorov, Petrovskii, Piskunov) equation. The solution uses a fractional generalization of the branching exponential process and propagation processes which are spectral integrals of Levy processes.

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{We deduce a shallow water model, describing the motion of the fluid in a lake, assuming inflow-outflow effects across the bottom. This model arises from the asymptotic analysis of the 3D dimensional Navier-Stokes equations. We prove the global in time existence result for this model in a bounded domain taking the nonlinear slip/friction boundary conditions to describe the inflows and outflows of the porous coast and the rivers. The solvability is shown in the class of solutions with L(p)-bounded vorticity for any given p is an element of (1, infinity). Copyright (C) 2009 John Wiley & Sons, Ltd.}

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{This work is concerned with the boundary layer turbulence, which is an outstanding problem in fluid mechanics. We consider an incompressible viscous fluid in 2D domains with permeable walls. The permeability is described by the Yudovich condition. The goal of the article is to study the fluid behavior at vanishing viscosity (large Reynold's numbers). We show that the vanishing viscous limit is a solution of the Euler equations with the Yudovich condition on the inflow region of the boundary. (C) 2013 Elsevier Ltd. All rights reserved.}

{We consider the Navier-Stokes equations in a 2D-bounded domain with general non-homogeneous Navier slip boundary conditions prescribed on permeable boundaries, and study the vanishing viscosity limit. We prove that solutions of the Navier-Stokes equations converge to solutions of the Euler equations satisfying the same Navier slip boundary condition on the inflow region of the boundary. The convergence is strong in Sobolev's spaces , which correspond to the spaces of the data.}

We study the inviscid limit for the two dimensional Navier-Stokes equations with non-homogeneous Navier slip boundary condition. We show that the vanishing viscosity limit of Navier-Stokes's solutions verifies the Euler equations with the corresponding Navier slip boundary condition just on the inflow boundary. The convergence result is established with respect to the strong topology of the Sobolev spaces W-p(1), p > 2.

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{We consider stochastic Navier-Stokes equations in a 2D-bounded domain with the Navier with friction boundary condition. We establish the existence and the uniqueness of the solutions and study the vanishing viscosity limit. More precisely, we prove that solutions of stochastic Navier Stokes equations converge, as the viscosity goes to zero, to solutions of the corresponding stochastic Euler equations. (C) 2015 Elsevier B.V. All rights reserved.}

{The theory of turbulent Newtonian fluids shows that the choice of the boundary condition is a relevant issue because it can modify the behavior of a fluid by creating or avoiding a strong boundary layer. In this study, we consider stochastic second grade fluids filling a two-dimensional bounded domain with the Navier-slip boundary condition (with friction). We prove the well-posedness of this problem and establish a stability result. Our stochastic model involves a multiplicative white noise and a convective term with third order derivatives, which significantly complicate the analysis. (C) 2017 Elsevier Inc. All rights reserved.}

{We consider a velocity tracking problem for the Navier-Stokes equations in a 2D bounded domain. The control acts on the boundary through an injection-suction device, and the flow is allowed to slip against the surface wall. We study the well-posedness of the state equations, linearized state equations, and adjoint equations. In addition, we show the existence of an optimal solution and establish the first order optimality condition.}

{We study the well-posedness of a stochastic differential equation on the two dimensional torus T-2, driven by an infinite dimensional Wiener process with drift in the Sobolev space L-2 (0, T; H-1 (T-2)). The solution corresponds to a stochastic Lagrangian flow in the sense of DiPerna Lions. By taking into account that the motion of a viscous incompressible fluid on the torus can be described through a suitable stochastic differential equation of the previous type, we study the inviscid limit. By establishing a large deviations principle, we show that, as the viscosity goes to zero, the Lagrangian stochastic Navier-Stokesflow approaches the Euler deterministic Lagrangian flow with an exponential rate function. (c) 2018 Elsevier Inc. All rights reserved.}