Global well-posedness of a system from quantum hydrodynamics for small data
Confluentes Mathematici, Tome 7 (2015) no. 2, pp. 7-16.

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 $0$, this condition is fullfilled thanks to recent scattering results.

DOI : https://doi.org/10.5802/cml.21
Classification : 35A01,  35Q31,  35Q55,  76D45
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Audiard, Corentin. Global well-posedness of a system from quantum hydrodynamics for small data. Confluentes Mathematici, Tome 7 (2015) no. 2, pp. 7-16. doi : 10.5802/cml.21. http://www.numdam.org/articles/10.5802/cml.21/

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