论文标题
开尔文波的稳定性和不稳定性
Stability and instability of Kelvin waves
论文作者
论文摘要
开尔文的$ m $ - 波动是2D Euler方程的均匀旋转贴片解决方案，$ m $ - 折叠的旋转对称性对于$ M \ geq 2 $。对于足够接近光盘的开尔文波，我们证明非线性稳定性结果在涡度的$ l^1 $范围内任意长时间，对于$ m $ $倍的对称扰动。为了获得这一结果，我们首先证明了开尔文波是某些可允许的斑块中能量功能的严格最大化，这是WAN在1986年所声称的。这给出了轨道稳定性的结果，具有扰动进化的支撑条件，但是使用Lagrangian Boottrap参数，可以限制脱落的范围，以置于侵蚀的范围内，我们可以在远离的范围内脱落。基于这种无条件的稳定性结果，我们确定在开尔文波附近发生了长时间的细丝或长臂的形成，这在各种数值模拟中都已观察到。此外，我们讨论了在同一变异框架中环形斑块的稳定性。
The $m$-waves of Kelvin are uniformly rotating patch solutions of the 2D Euler equations with $m$-fold rotational symmetry for $m\geq 2$. For Kelvin waves sufficiently close to the disc, we prove a nonlinear stability result up to an arbitrarily long time in the $L^1$ norm of the vorticity, for $m$-fold symmetric perturbations. To obtain this result, we first prove that the Kelvin wave is a strict local maximizer of the energy functional in some admissible class of patches, which had been claimed by Wan in 1986. This gives an orbital stability result with a support condition on the evolution of perturbations, but using a Lagrangian bootstrap argument which traces the particle trajectories of the perturbation, we are able to drop the condition on the evolution. Based on this unconditional stability result, we establish that long time filamentation, or formation of long arms, occurs near the Kelvin waves, which have been observed in various numerical simulations. Additionally, we discuss stability of annular patches in the same variational framework.
