Local hyperbolic structure of capillary water jet system

The instability of the capillary water jet system under long-wave perturbation—the Rayleigh-Plateau instability—has been observed and studied in experimental and theoretical physics since the 19th century. This presentation provides a rigorous mathematical justification for this phenomenon. We consider the capillary water jet system, modeled by the incompressible irrotational Euler equation with surface tension, and prove that it possesses a hyperbolic structure around the zero solution. The core of our method involves paradifferential calculus for conjugation in order to decouple this quasilinear system. This reduction enables the use of Lyapunov-Perron type arguments to construct the stable/unstable manifolds and a center invariant set. To our knowledge, this result is the first justification of (local) hyperbolic structure for a quasilinear system, offering a novel method potentially applicable to other quasilinear equations. This is a joint work with Chengyang Shao.