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Ergodicity Results for the Stochastic Navier–Stokes Equations: An Introduction

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Topics in Mathematical Fluid Mechanics

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2073))

Abstract

The theory of the stochastic Navier–Stokes equations (SNSE) has known a lot of important advances those last 20 years. Existence and uniqueness have been studied in various articles (see for instance [1, 35, 911, 13, 15, 21, 28, 30, 49, 51, 52, 66, 67]) and this part of the theory is well understood. Most of the deterministic results have been generalized to the stochastic context and it is now known that as in the deterministic case the SNSE has unique global strong solutions in dimension two. In dimension three, there exist global weak solutions and uniqueness is also a completely open problem in the stochastic case. The solutions in dimension three are weak in the sense of the theory of partial differential equations and in the sense of stochastic equations: the solutions are not smooth in space and they satisfy the SNSE only in the sense of the martingale problem. In Sect. 2 of these notes, we recall briefly these results and give the ideas of the proof.

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Notes

  1. 1.

    As is usual when deriving a priori estimates, many computations done in these notes are formal. Galerkin approximations can be used to prove rigorously that the final inequality is true.

  2. 2.

    Again, the following computation is formal and should be justified by Galerkin approximation for instance. In Sect. 4, we give details on the link between the Kolmogorov equation and the transition semi-group.

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    Arnaud Debussche

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Debussche, A. (2013). Ergodicity Results for the Stochastic Navier–Stokes Equations: An Introduction. In: Topics in Mathematical Fluid Mechanics. Lecture Notes in Mathematics(), vol 2073. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36297-2_2

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