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关于水动力内爆与兰道-库仑方程

On Hydrodynamic Implosions and the Landau-Coulomb Equation

The University of Chicago（芝加哥大学）；The University of Maryland（马里兰大学）

William Golding, Christopher Henderson

备注：
13 pages, Permanent preprint, see arXiv:2605.20426 instead
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AI总结：

本文研究了带有库仑势的非均匀兰道方程，并推导出一个新的延续准则：只要解保持有界，光滑解可以唯一地被延续。这提供了首个基于不控制质量密度量的延续准则，从而排除了因大$v$处衰减丧失而产生的内爆形成场景，即尾部变胖。此外，作者能够排除所有比类型I破裂速率更慢的类型II近似自相似破裂速率，无需对内侧轮廓做任何衰减假设，补充了现有文献中的类型I破裂分析。

摘要：

我们研究了带有库仑势的非均匀兰道方程，并推导出一个新的延续准则：只要解保持有界，光滑解可以唯一地被延续。这提供了，到我们所知为止，首个基于不控制质量密度量的延续准则。因此，我们能够排除一个已知的潜在奇点形成场景，即尾部变胖，其中由于大$v$处衰减的丧失而发生内爆。更一般地，我们能够排除所有比类型I破裂速率更慢的类型II近似自相似破裂速率，而无需对内侧轮廓做任何衰减假设，补充了现有文献中的类型I破裂分析。启发式地说，这表明应该不可能直接使用流体极限与三维可压缩欧拉方程的连接来构造带有库仑势的兰道方程的奇点解。这样的潜在内爆场景——基于三维欧拉方程的等熵或非等熵内爆——会自然导致一种缓慢的类型II近似自相似破裂场景，这完全落在我们定理的范围内。此预印本已被作者与路易斯·西尔维斯特雷合著的更近期工作“Pointwise bounds and obstructions to blowup for the Landau and Boltzmann equations”（arXiv:2605.20426）所吸收。此手稿将保持为永久预印本；所有引用应指向更近期的工作。

英文摘要：

We study the inhomogeneous Landau equation with Coulomb potential and derive a new continuation criterion: a smooth solution can be uniquely continued for as long as it remains bounded. This provides, to our knowledge, the first continuation criterion based on a quantity not controlling the mass density. Consequently, we are able to rule out a potential singularity formation scenario known as tail fattening, in which an implosion occurs due to the loss of decay at large $v$. More generally, we are able to rule out all Type II approximately self-similar blow-up rates that are slower than the Type I blow-up rate, without any assumption of decay on the inner profile, complementing existing Type I blow-up analysis in the literature. Heuristically, this suggests that it should be impossible to directly use the hydrodynamic limit connection with the 3D compressible Euler equations to construct a singular solution to the Landau equation with Coulomb potential. Such a potential implosion scenario -- based on either an isentropic or nonisentropic implosion for the 3D Euler equations -- would naturally result in a slow Type II approximately self-similar blow-up scenario, falling well within the range our theorem. This preprint has been subsumed by a more recent work by the authors and Luis Silvestre titled ``Pointwise bounds and obstructions to blowup for the Landau and Boltzmann equations,'' arXiv:2605.20426. This manuscript will remain a permanent preprint; all references should be directed to the more recent work.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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