Construction of High Regularity Invariant Measures for the 2D Euler Equations and Remarks on the Growth of the Solutions

Abstract

We consider the Euler equations on the torus in dimensions 2 and 3 and we construct invariant measures for the dynamics of these equations concentrated on sufficiently regular Sobolev spaces so that strong solutions are also known to exist at least locally. The proof follows the method of Kuksin in [17] and we obtain in particular that these measures do not have atoms, excluding trivial invariant measures such as diracs. Then we prove that µ-almost every initial data gives rise to a global solution for which the growth of the Sobolev norms are at most polynomial. We point out that up to the knowledge of the author, the only general upper-bound for the growth of the Sobolev norm to the 2d Euler equations is double exponential. In the other direction, some initial data are known to produce solutions exhibiting exponential growth, as in the work of Zlatos [26].

Almost sure global well-posedness for the energy supercritical NLS on the unit ball of $\mathbb{R}^3$

“…This is going to be combined with the local well-posedness result obtained in Theorem 3.1 and a globalization argument of Bourgain type (see the seminal work [5]). Our construction here has some important differences coming from the fact that our measures and not of Gibbs type, see also [53] and [43].…”

“…By considering a suitable modification (introducing a dissipation term and a carefully normalized white noise in time) of the equation, one then obtains that the stationary measure converges to a non-trivial invariant measure under the inviscid limit. See also [30,40,43,[51][52][53][54] for related works.…”

Global well-posedness and long-time behavior of the fractional NLS

“…This method is what we refer to Inviscid-Infinite-dimensional limit, or simply the "IID" limit. See also the work [40] for a similar procedure. We perform in Section 3 a general and independent version of the IID limit.…”