Year: 2026
Type: preprint
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)].
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171585256