一类流体力学模型的有限时间奇异性
Finite time singularities in a class of hydrodynamic models
摘要：
考虑了无粘不可压缩流体的模型，其中动能（即，拉格朗日泛函），在3D傅立叶表示中采用形式${\cal L}\sim\int k^α|{\bf v_k}|^2d^3{\bf k}$，其中$α$是常数，$0<α< 1$。与$α=0$（通常的欧拉流体力学）情况不同，$α$的有限值导致奇异冻结涡丝的有限能量。这个属性使我们能够研究这种细丝的动力学，而无需对短尺度进行正则化。对一对反平行涡丝进行了线性分析，发现在小波数时Crow不稳定性的模拟与静止解的小对称偏差有关。得到了该系统非线性长尺度动力学的局部近似哈密顿量。相应的方程的自相似解决方案被发现解析。它们描述了一个有限时间奇点的形成，所有的长度尺度都像$(t^*-t)^{1/(2-α)}$一样减小，其中$t^*$是奇点时间。
英文摘要：
Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form ${\cal L}\sim\int k^α|{\bf v_k}|^2d^3{\bf k}$ in 3D Fourier representation, where $α$ is a constant, $0<α< 1$. Unlike the case $α=0$ (the usual Eulerian hydrodynamics), a finite value of $α$ results in a finite energy for a singular, frozen-in vortex filament. This property allows us to study the dynamics of such filaments without the necessity of a regularization procedure for short length scales. The linear analysis of small symmetrical deviations from a stationary solution is performed for a pair of anti-parallel vortex filaments and an analog of the Crow instability is found at small wave-numbers. A local approximate Hamiltonian is obtained for the nonlinear long-scale dynamics of this system. Self-similar solutions of the corresponding equations are found analytically. They describe the formation of a finite time singularity, with all length scales decreasing like $(t^*-t)^{1/(2-α)}$, where $t^*$ is the singularity time.
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