一个受约束的log-HLS不等式及其相关的Wasserstein流
A constrained log-HLS inequality and its associated Wasserstein flow
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中文总结 AI 辅助
本文研究了受约束的log-HLS不等式及其诱导的Wasserstein流，该不等式与受约束的Trudinger-Moser-Onofri-Aubin不等式对偶，其梯度流导出PKS系统，约束将临界质量阈值提高两倍，从而提供了一种抑制爆破的新机制。
中文摘要 AI 辅助
我们研究了在约束条件下的log-Hardy-Littlewood-Sobolev（log-HLS）不等式以及由此自然诱导的Wasserstein流。该受约束的log-HLS不等式在变分意义下与受约束的Trudinger-Moser-Onofri-Aubin不等式是对偶的。相应的梯度流公式导出了一个抛物-椭圆型Patlak-Keller-Segel（PKS）系统，其全局动力学表现出依赖于初始质量的全局存在性与有限时间爆破之间的二分性。值得注意的是，该约束将流的临界质量阈值比无约束情形提高了两倍。这提供了一种与PKS系统相关公式中出现的抑制爆破不同的机制，在这些相关公式中，爆破通过附加效应（如强迫项、与流体方程的耦合或由Newtonian势的调和扰动产生的环境流）来缓解。
英文摘要
We study a log-Hardy-Littlewood-Sobolev (log-HLS) inequality under constraints and a naturally induced Wasserstein flow. The constrained log-HLS inequality is dual, in a variational sense, to the constrained Trudinger-Moser-Onofri-Aubin inequality. The corresponding gradient-flow formulation leads to a parabolic-elliptic Patlak-Keller-Segel (PKS) system whose global dynamics exhibit a dichotomy between global existence and finite-time blow-up depending on the initial mass. Remarkably, the constraint raises the critical mass threshold of the flow by a factor of two compared with the unconstrained setting. This provides a distinct mechanism for suppressing blow-up from those arising in related formulations of the PKS-system, where blow-up is mitigated through additional effects such as a forcing term, coupling with a fluid equation, or the ambient flow generated by harmonic perturbations of the Newtonian potential.
发表机构
- Tata Institute of Fundamental Research（塔塔基础研究所）
- Indian Statistical Institute（印度统计研究所）
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