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In this paper we prove that for all solutions of the 2d Euler equations withinitial vorticity with finite Sobolev smoothness then an initial data dependentnorm of the associated Lagrangian flow blows up in infinite time at least like$t^{\frac{1}{3}}$. This initial data dependent norm quantifies the exact $L^2$decay of the Fourier transform of the solution. This adapted norm turns out tobe the exact quantity that controls a low to high frequency cascade which wethen show to be the quantitative phenomenon behind the Lyapunov construction byShnirelman.

Small scale creation of the Lagrangian flow in 2d perfect fluids