Logarithmic Schr{ö}dinger equation and isothermal fluids

Abstract

We consider the large time behavior in two types of equations, posed on the whole space R d : the Schrödinger equation with a logarithmic nonlinearity on the one hand; compressible, isothermal, Euler, Korteweg and quantum Navier-Stokes equations on the other hand. We explain some connections between the two families of equations, and show how these connections may help having an insight in all cases. We insist on some specific aspects only, and refer to the cited articles for more details, and more complete statements. We try to give a general picture of the results, and present some heuristical arguments that can help the intuition, which are not necessarily found in the mentioned articles.

Qualitative analysis on logarithmic Schrödinger equation with general potential

“…For constant potential V , this equation was introduced in [4,5] to satisfy a specific tensorization property. See also [9] for a brief introduction on this property. The equation (1.2) has been applied in several fields such as the Bose-Einstein condensation, nuclear physics, and the optics.…”

“…The equation (1.2) has been applied in several fields such as the Bose-Einstein condensation, nuclear physics, and the optics. See [2,4,5,9,12,40] and the reference therein for more discussion on (1.2). We also refer [11] for the dispersive case.…”