On Shnirelman's inequality in 2D: Irreversible behavior and Euler-Lagrange variational formulation

Abstract: In this paper we prove a version of Shnirelman's inequality in the two-dimensional square $[0,1]<sup>2$. More precisely, given a target fluid configuration $f$ (a volume-preserving diffeomorphism) we construct a divergence-free velocity field $v\in L<sup>q_tL<sup>p_x$ whose flow connects $f$ to the identity and whose $L^q_{t} L<sup>p_x$-norm can effectively be bounded by the $L<sup>p$-difference of $f$ and $id$. However, higher regularity of the vector field might be affected. The main difference with the higher dimensional case is that, because of the topological obstructions of dimension $ν=2$, we have to allow that different fluid trajectories 'intersect' in space. We also observe that this choice leads to the emergence of irreversible behaviors in the Eulerian dynamics.