Li-Yau and Harnack estimates for nonlinear parabolic equations
Abstract
We establish global Harnack inequalities for positive viscosity solutions of several classes of nonlinear parabolic equations. The equations include positively homogeneous uniformly parabolic flows on Euclidean space and flat tori, normalized weighted p-Laplacian heat flows, and homogeneous isotropic diffusions on closed smooth metric measure spaces. For the normalized infinity-Laplacian heat flow, no curvature assumption is needed. For the ordinary weighted p-Laplacian, we obtain pressure estimates under nonnegative Bakry-Émery Ricci lower bounds. The proofs combine comparisons at two independent space-time points using the maximum principle for semicontinuous functions with weighted geodesic variations. The corresponding Li-Yau inequalities then follow in the first-order viscosity sense.
Related
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BibTeX
@misc{jeffres2026liyau,
title = {{Li-Yau and Harnack estimates for nonlinear parabolic equations}},
author = {Thalia Jeffres and Xiaolong Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09009},
eprint = {2610.09009},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09009},
}
RIS
TY - UNPB
TI - Li-Yau and Harnack estimates for nonlinear parabolic equations
AU - Thalia Jeffres
AU - Xiaolong Li
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.09009
UR - https://arxiv.org/abs/2610.09009
AB - We establish global Harnack inequalities for positive viscosity solutions of several classes of nonlinear parabolic equations. The equations include positively homogeneous uniformly parabolic flows on Euclidean space and flat tori, normalized weighted p-Laplacian heat flows, and homogeneous isotropic diffusions on closed smooth metric measure spaces. For the normalized infinity-Laplacian heat flow, no curvature assumption is needed. For the ordinary weighted p-Laplacian, we obtain pressure estimates under nonnegative Bakry-Émery Ricci lower bounds. The proofs combine comparisons at two independent space-time points using the maximum principle for semicontinuous functions with weighted geodesic variations. The corresponding Li-Yau inequalities then follow in the first-order viscosity sense.
ER -