no. 2
Global well-posedness of a system from quantum hydrodynamics for small data
Audiard, Corentin1
1 Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
Confluentes Mathematici, Tome 7 (2015) no. 2, pp. 7-17.

This article describes a joint work of the author with B.HaspotUmr Cnrs 7534, Université Paris Dauphine, place du Maréchal De Lattre De Tassigny 75775 Paris cedex 16 (France), haspot@ceremade.dauphine.fr 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.

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DOI : 10.5802/cml.21
Classification : 35A01, 35Q31, 35Q55, 76D45
Audiard, Corentin 1

1 Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France 
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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. 7-17. doi : 10.5802/cml.21. http://www.numdam.org/articles/10.5802/cml.21/

[1] Antonelli, Paolo; Marcati, Pierangelo On the finite energy weak solutions to a system in quantum fluid dynamics, Comm. Math. Phys., Volume 287 (2009) no. 2, pp. 657-686 | DOI

[2] Audiard, Corentin; Haspot, Boris From Gross-Pitaevskii equation to Euler-Korteweg system, existence of global strong solutions with small irrotational initial data, preprint

[3] Benzoni-Gavage, S.; Danchin, R.; Descombes, S. On the well-posedness for the Euler-Korteweg model in several space dimensions, Indiana Univ. Math. J., Volume 56 (2007), pp. 1499-1579

[4] Benzoni-Gavage, Sylvie; Danchin, Raphaël; Descombes, Stéphane Well-posedness of one-dimensional Korteweg models, Electron. J. Differential Equations (2006), No. 59, 35 pp. (electronic) pages

[5] Bona, J.; Ponce, G.; Saut, J.C.; Sparber, C. Dispersive blow up for nonlinear Schrödinger equations revisited, preprint

[6] Bourgain, J. Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., Volume 12 (1999) no. 1, pp. 145-171 | DOI

[7] Carles, Rémi; Danchin, Raphaël; Saut, Jean-Claude Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity, Volume 25 (2012) no. 10, pp. 2843-2873 | DOI

[8] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ℝ 3 , Ann. of Math. (2), Volume 167 (2008) no. 3, pp. 767-865 | DOI

[9] Gérard, P. The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 23 (2006) no. 5, pp. 765-779 | DOI

[10] Germain, P.; Masmoudi, N.; Shatah, J. Global solutions for the gravity water waves equation in dimension 3, Ann. of Math. (2), Volume 175 (2012) no. 2, pp. 691-754 | DOI

[11] Germain, Pierre; Masmoudi, Nader; Shatah, Jalal Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN (2009) no. 3, pp. 414-432 | DOI

[12] Ginibre, J.; Velo, G. Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9), Volume 64 (1985) no. 4, pp. 363-401

[13] Gustafson, Stephen; Nakanishi, Kenji; Tsai, Tai-Peng Scattering for the Gross-Pitaevskii equation, Math. Res. Lett., Volume 13 (2006) no. 2-3, pp. 273-285 | DOI

[14] Gustafson, Stephen; Nakanishi, Kenji; Tsai, Tai-Peng Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions, Ann. Henri Poincaré, Volume 8 (2007) no. 7, pp. 1303-1331 | DOI

[15] Gustafson, Stephen; Nakanishi, Kenji; Tsai, Tai-Peng Scattering theory for the Gross-Pitaevskii equation in three dimensions, Commun. Contemp. Math., Volume 11 (2009) no. 4, pp. 657-707 | DOI

[16] Hayashi, Nakao; Naumkin, Pavel I. On the quadratic nonlinear Schrödinger equation in three space dimensions, Internat. Math. Res. Notices (2000) no. 3, pp. 115-132 | DOI

[17] Kenig, Carlos E.; Merle, Frank Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., Volume 166 (2006) no. 3, pp. 645-675 | DOI

[18] Strauss, Walter Nonlinear Scattering Theory at Low Energy, J. Func. Anal., Volume 41 (1981), pp. 110-133

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