Quantified hydrodynamic limits for Schrödinger-type equations without the nonlinear potential

We establish rigorous and quantified hydrodynamic limits of the Schrödinger-type equations. Precisely, we consider the Schrödinger equation, Hartree equation, and Chern–Simons–Schrödinger equations and identify the hydrodynamic equations of them, when the Planck constant is negligible. In particular, we focus on the case when there is no Gross–Pitaevskii type self-interacting potential. In this situation, the method of modulated energy that has been used in previous literature to derive hydrodynamic limits of nonlinear Schrödinger-type equations is not valid anymore, since the system loses the desired estimate on the density difference. To overcome this difficulty, we combine the modulated energy with the bounded Lipschitz distance between densities to obtain the desired rigorous hydrodynamic limits of the Schrödinger-type equations without the potential.