粘性涡旋细丝的演化与孤子型传播
Evolution of viscous vortex filaments and soliton-type propagation
AI总结：
研究粘性不可压缩流体初始涡度在光滑曲线时的演化，通过特定构造及应用于桥本孤子，得出涡度主导阶描述及相关结论，包括动能分布局部部分在移动区域的特性等。
AI中文摘要：
我们研究了初始涡度支撑在光滑开放曲线上的粘性不可压缩流体的演化。结果表明，当$\nu t\ll 1$且雷诺数$\Gamma/\nu$足够小时，涡度的主导阶由集中在根据局部感应近似预测的副法线流演化的曲线上的兰姆 - 奥森型涡旋描述。解被写成显式主导阶轮廓加上低阶扰动，该扰动在莫雷$\mathcal{M}^{\infty}$范数下得到控制。然后，我们将此构造应用于桥本孤子。在此情况下，估计对于挠率参数是一致的，这使我们能够考虑孤子经历宏观位移的大挠率 regime。我们表明，相应的纳维 - 斯托克斯解包含与副法线速度相关的动能分布的局部部分，该部分在可允许的时间间隔内保持集中在移动的物理区域内并经历一阶位移。
英文摘要：
We study the evolution of a viscous incompressible fluid whose initial vorticity is supported on a smooth open curve. We show that, for $νt\ll 1$ and sufficiently small Reynolds number $Γ/ν$, the vorticity is described at leading order by a Lamb--Oseen type vortex concentrated around a curve evolving according to the binormal flow predicted by the localized induction approximation. The solution is written as an explicit leading-order profile plus a lower-order perturbation, which is controlled in a Morrey $\mathcal{M}^{\infty}$ norm. Then, we apply this construction to the Hasimoto soliton. In this case, the estimates are uniform with respect to the torsion parameter, allowing us to consider a large-torsion regime in which the soliton undergoes a macroscopic displacement. We show that the corresponding Navier--Stokes solution contains a localized portion of the kinetic-energy distribution, associated with the binormal velocity, which remains concentrated inside a moving physical region and undergoes an order-one displacement during an admissible time interval.
