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Some Methods of Infinite Dimensional Analysis in Hydrodynamics: An Introduction

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SPDE in Hydrodynamic: Recent Progress and Prospects

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1942))

In these lectures we shall concentrate on certain mathematical results concerning the case of deterministic Euler and stochastic Navier-Stokes equations for incompressible fluids. For general references we refer to [Fri95], [Tem83], [CF88], [VF88], [VKF79], [Lio96], [MP94], [NPe01], [Che96b], [Che98], [Che04], [CK04a], [Con95], [Con94], [Con01a], [Con01b], [FMRT01], [LR02], [MB02] [Tem84] and [Bir60] and for a discussion of challenging open problems see, e.g. [Fef06], [Can00], [Can04], [CF03], [Cho94], [Con95], [Con01a], [FMRT01], [Gal01], [ES00a], [Hey90], [Ros06] and [FMB03]. We shall concentrate particularly on the study of invariant measures associated with the above equations for fluids. On the one hand, this follows an analogy with the statistical mechanical approach to classical particle systems and ergodic theory, see, e.g. [Min00], [Rue69]. On the other hand, it follows Kolmogorov's suggestion, see e.g. [ER85], of adding small stochastic perturbations (“noise”) in classical dynamical systems, so to construct invariant measures and then study what happens when removing the noise.

The content of our lecture is as follows: in Section 2 we shall study the deterministic Euler equation and construct certain natural invariant measures for it. We also relate this analysis with the study of a certain Hamiltonian system describing vortices (“vortex models”). In Section 3 we shall study the stochastic Navier-Stokes equation with Gaussian space-time white noise and its invariant measure. We also provide brief comments and bibliographical references concerning recent work in directions which are complementary to those described here.

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  1. Institut för Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, 53115, Bonn, Germany

    Sergio Albeverio

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    Benedetta Ferrario

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    Michael Rückner

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Albeverio, S., Ferrario, B. (2008). Some Methods of Infinite Dimensional Analysis in Hydrodynamics: An Introduction. In: Da Prato, G., Rückner, M. (eds) SPDE in Hydrodynamic: Recent Progress and Prospects. Lecture Notes in Mathematics, vol 1942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78493-7_1

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