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A constrained log-HLS inequality and its associated Wasserstein flow

  • Monideep Ghosh
  • Debabrata Karmakar
  • Reetaj Sinha
  • Kartikey Verma
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Abstract

We study a log-Hardy-Littlewood-Sobolev (log-HLS) inequality under constraints and a naturally induced Wasserstein flow. The constrained log-HLS inequality is dual, in a variational sense, to the constrained Trudinger-Moser-Onofri-Aubin inequality. The corresponding gradient-flow formulation leads to a parabolic-elliptic Patlak-Keller-Segel (PKS) system whose global dynamics exhibit a dichotomy between global existence and finite-time blow-up depending on the initial mass. Remarkably, the constraint raises the critical mass threshold of the flow by a factor of two compared with the unconstrained setting. This provides a distinct mechanism for suppressing blow-up from those arising in related formulations of the PKS-system, where blow-up is mitigated through additional effects such as a forcing term, coupling with a fluid equation, or the ambient flow generated by harmonic perturbations of the Newtonian potential.

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Cite

BibTeXRIS
BibTeX
@misc{ghosh2026constrained,
  title = {{A constrained log-HLS inequality and its associated Wasserstein flow}},
  author = {Monideep Ghosh and Debabrata Karmakar and Reetaj Sinha and Kartikey Verma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05356},
  eprint = {2610.05356},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05356},
}
RIS
TY  - UNPB
TI  - A constrained log-HLS inequality and its associated Wasserstein flow
AU  - Monideep Ghosh
AU  - Debabrata Karmakar
AU  - Reetaj Sinha
AU  - Kartikey Verma
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.05356
UR  - https://arxiv.org/abs/2610.05356
AB  - We study a log-Hardy-Littlewood-Sobolev (log-HLS) inequality under constraints and a naturally induced Wasserstein flow. The constrained log-HLS inequality is dual, in a variational sense, to the constrained Trudinger-Moser-Onofri-Aubin inequality. The corresponding gradient-flow formulation leads to a parabolic-elliptic Patlak-Keller-Segel (PKS) system whose global dynamics exhibit a dichotomy between global existence and finite-time blow-up depending on the initial mass. Remarkably, the constraint raises the critical mass threshold of the flow by a factor of two compared with the unconstrained setting. This provides a distinct mechanism for suppressing blow-up from those arising in related formulations of the PKS-system, where blow-up is mitigated through additional effects such as a forcing term, coupling with a fluid equation, or the ambient flow generated by harmonic perturbations of the Newtonian potential.
ER  -