This article describes a joint work of the author with B.Haspot on the existence and uniqueness of global solutions for the Euler-Korteweg equations in the special case of quantum hydrodynamics. Our aim here is to sketch how one can construct global small solutions of the Gross-Pitaevskii equation and use the so-called Madelung transform to convert these into solutions without vacuum of the quantum hydrodynamics. A key point is to bound the the solution of the Gross-Pitaevskii equation away from , this condition is fullfilled thanks to recent scattering results.
@article{CML_2015__7_2_7_0,
author = {Audiard, Corentin},
title = {Global well-posedness of a system from quantum hydrodynamics for small data},
journal = {Confluentes Mathematici},
volume = {7},
year = {2015},
pages = {7-16},
doi = {10.5802/cml.21},
language = {en},
url = {http://dml.mathdoc.fr/item/CML_2015__7_2_7_0}
}
Audiard, Corentin. Global well-posedness of a system from quantum hydrodynamics for small data. Confluentes Mathematici, Tome 7 (2015) pp. 7-16. doi : 10.5802/cml.21. http://gdmltest.u-ga.fr/item/CML_2015__7_2_7_0/
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