对球对称多维无压欧拉-泊松系统尖锐阈值的完整刻画
A complete characterization of sharp thresholds to spherically symmetric multidimensional pressureless Euler-Poisson systems
AI总结：
本文研究了球对称多维无压欧拉-泊松系统中，通过建立尖锐临界阈值条件，区分全局正则性与有限时间破裂，揭示了关键分析方法和统一处理框架。
摘要：
欧拉-泊松（EP）系统模型了多种物理过程的动力学，包括电荷输运、碰撞等离子体以及某些宇宙波现象。在本工作中，我们建立了区分全局时间正则性与有限时间破裂的EP系统径向对称、多维无压系统的尖锐临界阈值条件。总体而言，有两种情况：有背景（c>0）和无背景（c=0）。对于c>0，我们假设周期性条件获得精确阈值。我们方法的一个关键特征是能够无缝扩展到零背景情况，其中我们获得尖锐阈值而不需任何额外假设。特别是，该框架允许初始速度可能为负，使流体流向原点。EP系统导出阈值条件的主要分析挑战源于各种局部/非局部力的复杂耦合。为克服这一挑战，我们识别了一个新的非线性量，在分析中起决定性作用，并使所有相关场景的统一处理成为可能。我们的结果为多维无压EP系统的临界阈值提供了全面的刻画。
英文摘要：
The Euler-Poisson (EP) system models the dynamics of a variety of physical processes, including charge transport, collisional plasmas, and certain cosmological wave phenomena. In this work, we establish sharp critical threshold conditions that distinguish global-in-time regularity from finite-time breakdown for solutions of the radially symmetric, multidimensional pressureless EP system. Overall, there are two cases: with and without background ($c>0, c=0$ respectively). For $c>0$, we obtain precise thresholds assuming a periodicity condition. A key feature of our approach is that it extends seamlessly to the zero background case, where we obtain sharp thresholds without imposing any additional assumptions. In particular, the framework accommodates initial velocities that may be negative, allowing the flow to be directed toward the origin. The main analytical challenge of deriving threshold conditions for EP systems stems from the intricate coupling of various local/nonlocal forces. To overcome this, we identify a novel nonlinear quantity that plays a decisive role in the analysis and enables a unified treatment of all relevant scenarios. Our results provide a comprehensive characterization of critical thresholds for the pressureless EP system in multiple dimensions.
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