Blow up and Concentration without Quantization: sharp Harnack type inequalities
Abstract
Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we refine the blow up analysis for sequences of solutions of a class of perturbed singular Liouville equations which share the phenomenon of ``blow up and concentration without quantization". The problem is delicate because we are dealing with the exact threshold value above which one meets the well known ``concentration without quantization" phenomenon, as recently pushed forward in [C.S. Lin, G. Tarantello, {\em C. R. Math. Acad. Sci. Paris} (2016)] and in [Y. Lee, C.S. Lin, G. Tarantello, W. Yang, {\em Comm. PDE.} (2017)]. We overcome these difficulties by developing different strategies for ``fast" and ``slow" blow up, combining refined Pohozaev identities, elliptic estimates and ``Sup+CInf" inequalities in the same spirit of [C.C. Chen, C.S. Lin, {\em Comm. An. Geom.} (1998)]. In the process, we also derive a new sharp Harnack type inequality by using a moving-plane method, following the ideas in [D. Bartolucci, C.C. Chen, C.S. Lin, G. Tarantello, {\em Comm. P.D.E.} (2002)].
Related
- Lorentzian entropy, subgaussian matrix concentration, and sharp functional inequalities for all matroids
- Harnack-type inequalities for nonlocal doubly nonlinear parabolic equations
- Characterization of intrinsic parabolic Harnack inequalities for doubly nonlinear equations of slow diffusion type
- Differential Harnack inequalities and maximum principles of Morel-Oswald type for elliptic PDE in divergence form
- A Sharp Gradient Harnack Estimate for the Lane--Emden System
BibTeX
@misc{bartolucci2026blow,
title = {{Blow up and Concentration without Quantization: sharp Harnack type inequalities}},
author = {Daniele Bartolucci and Paolo Cosentino and Lina Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.24346},
eprint = {2607.24346},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.24346},
}
RIS
TY - UNPB
TI - Blow up and Concentration without Quantization: sharp Harnack type inequalities
AU - Daniele Bartolucci
AU - Paolo Cosentino
AU - Lina Wu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2607.24346
UR - https://arxiv.org/abs/2607.24346
AB - Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we refine the blow up analysis for sequences of solutions of a class of perturbed singular Liouville equations which share the phenomenon of ``blow up and concentration without quantization". The problem is delicate because we are dealing with the exact threshold value above which one meets the well known ``concentration without quantization" phenomenon, as recently pushed forward in [C.S. Lin, G. Tarantello, {\em C. R. Math. Acad. Sci. Paris} (2016)] and in [Y. Lee, C.S. Lin, G. Tarantello, W. Yang, {\em Comm. PDE.} (2017)]. We overcome these difficulties by developing different strategies for ``fast" and ``slow" blow up, combining refined Pohozaev identities, elliptic estimates and ``Sup+CInf" inequalities in the same spirit of [C.C. Chen, C.S. Lin, {\em Comm. An. Geom.} (1998)]. In the process, we also derive a new sharp Harnack type inequality by using a moving-plane method, following the ideas in [D. Bartolucci, C.C. Chen, C.S. Lin, G. Tarantello, {\em Comm. P.D.E.} (2002)].
ER -