Stability for the quadratic derivative nonlinear Schrödinger equation and applications to the Korteweg–Kirchhoff type Euler equations for quantum hydrodynamics

This paper is concerned with an existence and stability result on the nonlinear derivative Schrödinger equation in 1-D, which is originated by the study of the stability of nontrivial steady states in Quantum Hydrodynamics. The problem is equivalent to a compressible Euler fluid system with a very specific Korteweg–Kirchhoff stress K(ρ)=ħ4ρ. As a simple, but significative, example we consider the nonlinear derivative Schrödinger equation obtained via a complex Cole–Hopf type transformation, applied to the 1-D free Schrödinger equation. The resulting problem (possibly unstable) is investigated for small solutions around the null steady state. The stability is proved to be valid for long time intervals of order O(ϵ−4∕5), where ϵ is the order of smallness of the initial data. This result brought back to the QHD system provides the stability of the steady state ρ=1,J=v=0. The validity in time of this result is far beyond what can be obtained via classical linearization analysis or via higher order energy estimates. Indeed in our analysis the nonlinear structure plays a crucial role in the corresponding iteration procedure, the use of local smoothing and the Schrödinger maximal operator provides the control of the potential lost of regularity.