Global existence and convergence of a density heat flow for strictly subcritical Liouville systems
Abstract
We study a density heat flow associated with normalized Liouville systems on a closed connected Riemannian surface, and we show that this flow leads to the existence of solutions for the corresponding subcritical Liouville systems. The coupling matrix is symmetric and entrywise nonnegative, with no invertibility, irreducibility or inverse-matrix sign assumption. Under the strict subcriticality conditions 8π∑i∈ Jρi>∑i,j∈ Jaijρiρj for every nonempty set of components J, every smooth positive initial density of masses ρi>0 generates a unique global smooth positive solution converging in every Cm norm to a stationary density. The limiting potentials solve the stationary Liouville system. Our proof uses entropy coercivity and a gradient inequality in the density variables, which also yields algebraic or exponential convergence rates.
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BibTeX
@misc{cheng2026global,
title = {{Global existence and convergence of a density heat flow for strictly subcritical Liouville systems}},
author = {Zetao Cheng and Haoyu Li and Woongbae Park and Lei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14455},
eprint = {2610.14455},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14455},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global existence and convergence of a density heat flow for strictly subcritical Liouville systems
AU - Zetao Cheng
AU - Haoyu Li
AU - Woongbae Park
AU - Lei Zhang
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14455
UR - https://arxiv.org/abs/2610.14455
AB - We study a density heat flow associated with normalized Liouville systems on a closed connected Riemannian surface, and we show that this flow leads to the existence of solutions for the corresponding subcritical Liouville systems. The coupling matrix is symmetric and entrywise nonnegative, with no invertibility, irreducibility or inverse-matrix sign assumption. Under the strict subcriticality conditions 8π∑_i∈ Jρ_i>∑_i,j∈ Ja_ijρ_iρ_j for every nonempty set of components J, every smooth positive initial density of masses ρ_i>0 generates a unique global smooth positive solution converging in every C^m norm to a stationary density. The limiting potentials solve the stationary Liouville system. Our proof uses entropy coercivity and a gradient inequality in the density variables, which also yields algebraic or exponential convergence rates.
ER -