论文标题
对称双重周期性重力 - 毛细血管波和涡旋较小
Symmetric doubly periodic gravity-capillary waves with small vorticity
论文作者
论文摘要
我们在三个空间尺寸中构造了具有小非零涡度的小重力 - 毛细血管，并从均匀的流中分叉。波是对称的，并且在两个水平坐标中周期性。该证明是由洛尔茨在反射对称环形域中构造磁液静态平衡的启发。它依赖于涡度的全球表示作为两个梯度的跨产品，并规定了伯努利函数与水颗粒的轨道周期之间的功能关系。自由表面的存在引入了重大的新挑战。特别是，由此产生的自由边界问题不是椭圆形的，并且所涉及的地图在Fréchet分化下会丧失规律性。但是，我们表明，通过仔细跟踪规律性丧失，适用了Crandall-Rabinowitz本地分叉方法的版本。
We construct small amplitude gravity-capillary water waves with small nonzero vorticity, in three spatial dimensions, bifurcating from uniform flows. The waves are symmetric, and periodic in both horizontal coordinates. The proof is inspired by Lortz' construction of magnetohydrostatic equilibria in reflection-symmetric toroidal domains. It relies on a global representation of the vorticity as the cross product of two gradients, and on prescribing a functional relationship between the Bernoulli function and the orbital period of the water particles. The presence of the free surface introduces significant new challenges. In particular, the resulting free boundary problem is not elliptic, and the involved maps incur a loss of regularity under Fréchet differentiation. Nevertheless, we show that a version of the Crandall-Rabinowitz local bifurcation method applies, by carefully tracking the loss of regularity.
