Logarithmic Schr{\"o}dinger equation and isothermal fluids

DOI10.4171/EMSS/54arXiv2108.13021WikidataQ113692018 ScholiaQ113692018MaRDI QIDQ6376310

Publication date: 30 August 2021

Abstract: We consider the large time behavior in two types of equations, posed on the whole space R^d: the Schr{"o}dinger 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.

Mathematics Subject Classification ID

Asymptotic behavior of solutions to PDEs (35B40) PDEs in connection with fluid mechanics (35Q35) NLS equations (nonlinear Schrödinger equations) (35Q55) Quantum hydrodynamics and relativistic hydrodynamics (76Y05) Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics (76N10) PDEs in connection with classical thermodynamics and heat transfer (35Q79) Time-dependent Schrödinger equations and Dirac equations (35Q41) Euler equations (35Q31)

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