On Shnirelman's inequality in 2D: Irreversible behavior and Euler-Lagrange variational formulation
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Schiffer, Stefan, Zizza, Martina
Institution: Max Planck Institute for Mathematics in the Sciences
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214659545