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PreprintarXiv licencearXiv

Inverse Energy Cascade and Non-uniqueness for the Advection-Diffusion Equation

  • Alexey Cheskidov
  • Xiaoran Liu
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Abstract

We prove non-uniqueness for the advection-diffusion equation with a drift in a logarithmically near-critical class below Lt2Lx∞. The first construction is a diffusion-assisted inverse cascade of scalar energy that diverges as t → 0. The second uses compressed inverse mixing, with diffusion acting perturbatively, and produces a bounded parabolic solution with the energy jump at the initial time. In both constructions the drift and scalar are smooth for positive time. We also consider the drift in L2,∞tL∞x and solutions satisfying the energy inequality starting from almost every time. In this setting, we prove uniqueness of solutions with bounded energy, and construct non-unique solution with energy blowing up as time goes to zero.

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Cite

BibTeXRIS
BibTeX
@misc{cheskidov2026inverse,
  title = {{Inverse Energy Cascade and Non-uniqueness for the Advection-Diffusion Equation}},
  author = {Alexey Cheskidov and Xiaoran Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01779},
  eprint = {2610.01779},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01779},
}
RIS
TY  - UNPB
TI  - Inverse Energy Cascade and Non-uniqueness for the Advection-Diffusion Equation
AU  - Alexey Cheskidov
AU  - Xiaoran Liu
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01779
UR  - https://arxiv.org/abs/2610.01779
AB  - We prove non-uniqueness for the advection-diffusion equation with a drift in a logarithmically near-critical class below L_t^2L_x^∞. The first construction is a diffusion-assisted inverse cascade of scalar energy that diverges as t → 0. The second uses compressed inverse mixing, with diffusion acting perturbatively, and produces a bounded parabolic solution with the energy jump at the initial time. In both constructions the drift and scalar are smooth for positive time. We also consider the drift in L^2,∞_tL^∞_x and solutions satisfying the energy inequality starting from almost every time. In this setting, we prove uniqueness of solutions with bounded energy, and construct non-unique solution with energy blowing up as time goes to zero.
ER  -