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arXiv · 2609.31294

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

Stefan Schiffer·Martina Zizza
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Abstract

In this paper we prove a version of Shnirelman's inequality in the two-dimensional square $[0,1]^2$. More precisely, given a target fluid configuration $f$ (a volume-preserving diffeomorphism) we construct a divergence-free velocity field $v\in L^q_tL^p_x$ whose flow connects $f$ to the identity and whose $L^q_{t} L^p_x$-norm can effectively be bounded by the $L^p$-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.

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Stefan Schiffer, Martina Zizza. 2026-09-25. On Shnirelman's inequality in 2D: Irreversible behavior and Euler-Lagrange variational formulation. https://arxiv.org/abs/2609.31294

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