流体动力学流动中爆破的重整化和普适性
Renormalization and universality of blowup in hydrodynamic flows
摘要：
我们考虑描述湍流壳模型中间歇性爆发的自相似解，并研究它们与连续流体力学模型中爆破现象的关系。首先，我们表明，这些解决方案是非常接近的自相似解的傅立叶变换的无粘Burgers方程对应的冲击形成从光滑的初始数据。然后，将所得结果推广到一维可压缩流的双曲型守恒律。结果表明，重整化后的波剖面趋于一个普适函数，它既与初始条件无关，也与特定形式的守恒定律无关。这种现象可以看作是重整化群理论的一种新的表现形式。最后，我们讨论了应用开发的理论检测和描述在不可压缩流爆破的可能性。
英文摘要：
We consider self-similar solutions describing intermittent bursts in shell models of turbulence, and study their relationship with blowup phenomena in continuous hydrodynamic models. First, we show that these solutions are very close to self-similar solution for the Fourier transformed inviscid Burgers equation corresponding to shock formation from smooth initial data. Then, the result is generalized to hyperbolic conservation laws in one space dimension describing compressible flows. It is shown that the renormalized wave profile tends to a universal function, which is independent both of initial conditions and of a specific form of the conservation law. This phenomenon can be viewed as a new manifestation of the renormalization group theory. Finally, we discuss possibilities for application of the developed theory for detecting and describing a blowup in incompressible flows.
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