A rigidity result for the 3D capillary liquid drop with constant vorticity

We consider the free boundary problem for the Euler equations of fluid dynamics governing the motion of a 3D liquid drop with capillarity $\sigma_0$ and nearly spherical shape, under the assumption of constant vorticity $(0, 0, \alpha_0)$.

First we study the compatibility of the constant vorticity condition with the evolution in time of the system, showing that, for $\alpha_0 \neq 0$, any smooth solution with convex domain must satisfy a strong geometrical constraint on the shape of the fluid domain, and that the constant vorticity condition (unlike in the irrotational case $\alpha_0 = 0$) does not define an invariant set for the time evolution of the system.

Then we focus on the time-independent solutions of the problem and we prove a new rigidity result: starting without assuming any symmetry condition, we show that, if the ratio $\alpha_0^2/\sigma_0$ is not too large, then any nearly spherical solution has necessarily cylindrical symmetry, and therefore it is the unique axisymmetric solution already known in literature, the fluid domain is close, but not equal, to a ball, more precisely it is an oblate spheroid, flattened at the poles and bulged at the equator, and each fluid particle moves along a horizontal, circular trajectory with constant angular velocity.

To the best of our knowledge, this is the first result for the capillary liquid drop with constant vorticity obtained without assuming cylindrical symmetry.