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arXiv · 2610.05356

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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Monideep Ghosh, Debabrata Karmakar, Reetaj Sinha, Kartikey Verma. 2026-10-04. A constrained log-HLS inequality and its associated Wasserstein flow. https://arxiv.org/abs/2610.05356

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