Local Invariant Structures in the Dynamics of Capillary Water Jet

Abstract

Physical experiments show that a capillary water jet is exponentially unstable under long wave perturbations, while remaining stable under short wave perturbations. Measurements indicate that the exponential growth rate in the long wave regime agrees quantitatively with the classical predictions of Rayleigh and Plateau, known as \emph{Rayleigh-Plateau instability}. In this paper, we provide a mathematical justification of these experimental observations. The motion of the water jet is modeled by an axisymmetric irrotational Eulerian free-boundary system governed by surface tension. We prove that the (un)stable directions in the linearized system, corresponding to long wave perturbations, are indeed tangent to a (un)stable invariant manifold of the full nonlinear system. On the other hand, the elliptic directions, corresponding to short wave perturbations, are indeed tangent to a center invariant set in a generalized sense. These results give a positive answer to the question raised by Lin-Zeng concerning the existence of invariant manifolds for Eulerian free-boundary systems. A notable finding is the existence of infinite dimensional hyperbolic invariant manifolds in the dynamics of infinitely long capillary water jet, in absence of spectral gap. The major methodological contribution is the construction of ``paradifferential propagator" corresponding to linear paradifferential hyperbolic systems, effectively balancing the loss of regularity due to quasilinearity. The method can be generalized to a broad class of quasilinear problems.