Navier-Stokes爆破的级联机制
Cascade mechanisms for Navier-Stokes blow-up
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- Westlake University（西湖大学）
- University of Illinois at Chicago（伊利诺伊大学芝加哥分校）
- Princeton University（普林斯顿大学）
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中文总结 AI 辅助
本文通过Obukhov二进模型阐明三维Navier-Stokes方程的逆能量级联导致瞬时I型爆破的机制，并证明在高间歇性下有限能量解中同样发生，同时给出混合模型的有限时间爆破定理。
中文摘要 AI 辅助
在最近的一篇预印本（arXiv:2511.09556）中，我们展示了三维Navier-Stokes方程存在逆能量级联，该级联在特定尖锐正则性类中导致“瞬时”I型爆破以及唯一性的失效。本文旨在Obukhov二进模型背景下进一步阐明这一现象，并将其与更为熟知的有限时间爆破现象进行比较。当间歇性较低（$\alpha\leq 2$）时，我们恢复了之前在Navier-Stokes情形下的结果；在能量超临界且间歇性较高（$\alpha>2$）的情形下，我们证明同样的情况可以发生在有限能量解类中。我们给出了两种不同的证明：一种使用Lyapunov函数的软性方法，以及一种显式的多尺度构造。对于无粘系统，我们说明了类似现象会在较低正则性下发生。最后，我们陈述了一个针对混合Desnyansky-Novikov-Obukhov模型的有限时间爆破定理，在该模型中，来自有限支撑数据且无外力的前向级联产生了一个仅以微小裕度为II型的奇点，其特征与OpenAI最近的受迫Navier-Stokes爆破相当。
英文摘要
In a recent preprint (arXiv:2511.09556), we exhibited an inverse energy cascade for the 3D Navier-Stokes equations which results in "instantaneous" Type I blow-up and failure of uniqueness in certain sharp regularity classes. Our purpose here is to elucidate this phenomenon further in the setting of the Obukhov dyadic model, and to compare it to the better-known phenomenon of finite-time blow-up. When intermittency is low ($α\leq 2$), we recover our previous result from the Navier-Stokes setting; in the energy-supercritical case where intermittency is high ($α>2$), we show that the same can occur in the class of finite energy solutions. We present two different proofs: a soft approach using a Lyapunov function and an explicit multiscale construction. For the inviscid system we illustrate that a similar phenomenon occurs at lower regularity. Finally, we state a finite-time blow-up theorem for a mixed Desnyansky-Novikov-Obukhov model, in which a forward cascade from finitely supported data and no force produces a singularity that is of Type II only by a small margin, with features comparable to the recent forced Navier-Stokes blow-up of OpenAI.
