Long time strong Sobolev instability for quasilinear NLS

Abstract

We prove long-time strong Sobolev instability for a class of quasilinear Schr\"odinger equations on $\mathbb T^2$ arising, via the Madelung transform, in the description of Euler-Korteweg capillary fluids. More precisely, for every $s\ge 7$, arbitrarily small $\mu>0$, and arbitrarily large $\mathcal K>0$, we construct a solution such that $$ \|u(0)\|_{H^s}<\mu, \qquad \|u(T)\|_{H^s}>\mathcal K $$ at some finite time $T$, for which we provide an explicit exponential upper bound. The instability mechanism is generated by a finite-dimensional resonant Toy Model associated with the cubic semilinear part of the equation, in the spirit of Colliander-Keel-Staffilani-Takaoka-Tao. The main technical difficulty is to justify this dynamics over the long time scale of the energy transfer in the presence of quasilinear derivative losses. We achieve this by combining a weak Birkhoff normal form with paradifferential calculus, microlocal symmetrization, and modified energy estimates. Our result provides a rigorous quasilinear realization of a resonant energy-transfer mechanism, and suggests a connection with weakly turbulent dynamics in capillary fluids.