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PreprintCC BY 4.0arXiv

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∈ LqtLpx whose flow connects f to the identity and whose Lqt Lpx-norm can effectively be bounded by the Lp-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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Cite

BibTeXRIS
BibTeX
@misc{schiffer2026shnirelmans,
  title = {{On Shnirelman's inequality in 2D: Irreversible behavior and Euler-Lagrange variational formulation}},
  author = {Stefan Schiffer and Martina Zizza},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31294},
  eprint = {2609.31294},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31294},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Shnirelman's inequality in 2D: Irreversible behavior and Euler-Lagrange variational formulation
AU  - Stefan Schiffer
AU  - Martina Zizza
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31294
UR  - https://arxiv.org/abs/2609.31294
AB  - 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∈ 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.
ER  -