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We consider the Euler equations on the two-dimensional torus and constructinvariant measures for the dynamics of these equations, concentrated onsufficiently regular Sobolev spaces so that strong solutions are also known toexist. The proof follows the method of Kuksin and we obtain in particular thatthese measures do not have atoms, excluding trivial invariant measures. Then weprove that almost every initial data with respect to the constructed measuresgive rise to global solutions for which the growth of the Sobolev norms are atmost polynomial. To do this, we rely on an argument of Bourgain. Such acombination of Kuksin's and Bourgain's arguments already appears in the work ofSy. We point out that up to the knowledge of the author, the only general upperbound for the growth of the Sobolev norm to the $2d$ Euler equations is doubleexponential.

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