无量化的爆破与集中：尖锐的哈拿克型不等式
Blow up and Concentration without Quantization: sharp Harnack type inequalities
浏览论文内容
中文总结 AI 辅助
受湍流欧拉流启发，改进一类扰动奇异刘维尔方程解序列的爆破分析，针对‘无量化的爆破与集中’现象，先提出新的尖锐哈拿克型不等式，又通过不同策略解决奇点共形因子增长带来的问题。
中文摘要 AI 辅助
受具有点奇点的湍流欧拉流的昂萨格统计力学描述的启发，我们改进了一类扰动奇异刘维尔方程解序列的爆破分析，这类方程存在‘无量化的爆破与集中’现象。该问题很微妙，因为我们要处理恰好高于此阈值就会出现‘无量化集中’现象的情况。首先，对于这个特别丰富的奇异极限，我们需要一个新的尖锐哈拿克型不等式。然而这还不够，由于奇点继承的共形因子的增长阻碍了经典量化论证的使用。我们还通过基于波霍扎耶夫恒等式、椭圆估计和‘上确界 + 下确界’不等式的不同策略，针对‘快速’和‘缓慢’爆破解决了这个问题。
英文摘要
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, C. R. Math. Acad. Sci. Paris (2016)] and in [Y. Lee, C.S. Lin, G. Tarantello, W. Yang, Comm. PDE. (2017)]. First of all we need a new sharp Harnack type inequality for this particularly rich singular limit. However this is not enough, since the growth of the conformal factor inherited by the singularity prevents the use of classical quantization arguments. We solve also this issue with different strategies for "fast" and "slow" blow up, by a careful adaptation of arguments based on the Pohozaev identity, elliptic estimates and "Sup+CInf" inequalities in the same spirit of [C.C. Chen, C.S. Lin, Comm. An. Geom. (1998)].
