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This volume deals with the random perturbation of PDEs which lack well-posedness, mainly because of their non-uniqueness, in some cases because of blow-up. The aim is to show that noise may restore uniqueness or prevent blow-up. This is not a general or easy-to-apply rule, and the theory presented in the book is in fact a series of examples with a few unifying ideas. The role of additive and bilinear multiplicative noise is described and a variety of examples are included, from abstract parabolic evolution equations with non-Lipschitz nonlinearities to particular fluid dynamic models, like the dyadic model, linear transport equations and motion of point vortices.

Perturbation (Mathematics) --- Fluid dynamics --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Mathematical Statistics --- Applied Mathematics --- Mathematical models --- Fluid dynamic measurements. --- Differential equations, Partial. --- Partial differential equations --- Measurements, Fluid dynamic --- Perturbation equations --- Perturbation theory --- Measurement --- Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Physical measurements --- Approximation theory --- Dynamics --- Functional analysis --- Mathematical physics --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities

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This book is devoted to stochastic Navier–Stokes equations and more generally to stochasticity in fluid mechanics. The two opening chapters describe basic material about the existence and uniqueness of solutions: first in the case of additive noise treated pathwise and then in the case of state-dependent noise. The main mathematical techniques of these two chapters are known and given in detail for using the book as a reference for advanced courses. By contrast, the third and fourth chapters describe new material that has been developed in very recent years or in works now in preparation. The new material deals with transport-type noise, its origin, and its consequences on dissipation and well-posedness properties. Finally, the last chapter is devoted to the physical intuition behind the stochastic modeling presented in the book, giving great attention to the question of the origin of noise in connection with small-scale turbulence, its mathematical form, and its consequences on large-scale properties of a fluid.

Differential equations. --- Stochastic analysis. --- Functional analysis. --- Differential Equations. --- Stochastic Analysis. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analysis, Stochastic --- Mathematical analysis --- Stochastic processes --- 517.91 Differential equations --- Differential equations --- Fluid mechanics. --- Stochastic partial differential equations. --- Mecànica de fluids --- Equacions diferencials parcials estocàstiques --- Equacions de Navier-Stokes

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This book presents in thirteen refereed survey articles an overview of modern activity in stochastic analysis, written by leading international experts. The topics addressed include stochastic fluid dynamics and regularization by noise of deterministic dynamical systems; stochastic partial differential equations driven by Gaussian or Lévy noise, including the relationship between parabolic equations and particle systems, and wave equations in a geometric framework; Malliavin calculus and applications to stochastic numerics; stochastic integration in Banach spaces; porous media-type equations; stochastic deformations of classical mechanics and Feynman integrals and stochastic differential equations with reflection. The articles are based on short courses given at the Centre Interfacultaire Bernoulli of the Ecole Polytechnique Fédérale de Lausanne, Switzerland, from January to June 2012. They offer a valuable resource not only for specialists, but also for other researchers and Ph.D. students in the fields of stochastic analysis and mathematical physics. Contributors: S. Albeverio M. Arnaudon V. Bally V. Barbu H. Bessaih Z. Brzeźniak K. Burdzy A.B. Cruzeiro F. Flandoli A. Kohatsu-Higa S. Mazzucchi C. Mueller J. van Neerven M. Ondreját S. Peszat M. Veraar L. Weis J.-C. Zambrini.

Partial differential equations --- Operational research. Game theory --- Probability theory --- Mathematics --- Applied physical engineering --- differentiaalvergelijkingen --- toegepaste wiskunde --- waarschijnlijkheidstheorie --- stochastische analyse --- economie --- wiskunde --- kansrekening

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Of the three lecture courses making up the CIME summer school on Fluid Dynamics at Cetraro in 2005 reflected in this volume, the first, due to Sergio Albeverio describes deterministic and stochastic models of hydrodynamics. In the second course, Franco Flandoli starts from 3D Navier-Stokes equations and ends with turbulence. Finally,Yakov Sinai, in the 3rd course, describes some rigorous mathematical results for multidimensional Navier-Stokes systems and some recent results on the one-dimensional Burgers equation with random forcing.

Partial differential equations --- Operational research. Game theory --- Mathematical physics --- differentiaalvergelijkingen --- stochastische analyse --- wiskunde --- fysica --- kansrekening