一维流体动力学方程热化的开始：Burgers方程的启示
The Onset of Thermalisation in Finite-Dimensional Equations of Hydrodynamics: Insights from the Burgers Equation
摘要：
有限维（所有空间傅立叶模式设置为零，超过一个有限波数$K_G$），无粘流体动力学方程的解决方案，在很长的时间是已知的差异与原来的无限维偏微分方程或其粘性对应。令人惊讶的是，这种伽辽金截断方程的解发展出尖锐的局部结构，称为{ \it tygers} [Ray等人，E { \bf 84}，016301（2011）]，其最终导致与均分能谱相关联的完全热化状态。现在，我们得到，通过使用解析易处理的伯格斯方程，精确的估计，理论上和通过直接的数值模拟，在该时间$τ_c$热化被触发，并显示$τ_c \sim \kg^ξ$，与$ξ= -4/9$。我们的研究结果有几个方面的影响，包括解析条方法数值获得证据或反对爆破的三维不可压缩欧拉方程。
英文摘要：
Solutions to finite-dimensional (all spatial Fourier modes set to zero beyond a finite wavenumber $K_G$), inviscid equations of hydrodynamics at long times are known to be at variance with those obtained for the original infinite dimensional partial differential equations or their viscous counterparts. Surprisingly, the solutions to such Galerkin-truncated equations develop sharp localised structures, called {\it tygers} [Ray, et al., Phys. Rev. E {\bf 84}, 016301 (2011)], which eventually lead to completely thermalised states associated with an equipartition energy spectrum. We now obtain, by using the analytically tractable Burgers equation, precise estimates, theoretically and via direct numerical simulations, of the time $τ_c$ at which thermalisation is triggered and show that $τ_c \sim \kg^ξ$, with $ξ= -4/9$. Our results have several implications including for the analyticity strip method to numerically obtain evidence for or against blow-ups of the three-dimensional incompressible Euler equations.
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