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/

[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 | Article | MR 2481754 | Zbl 1177.82127

[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 | MR 2354691 | Zbl 1125.76060

[4] Benzoni-Gavage, Sylvie; Danchin, Raphaël; Descombes, Stéphane Well-posedness of one-dimensional Korteweg models, Electron. J. Differential Equations (2006), pp. No. 59, 35 pp. (electronic) | MR 2226932 | Zbl 1114.76058

[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 | Article | MR 1626257 | Zbl 0958.35126

[7] Carles, Rémi; Danchin, Raphaël; Saut, Jean-Claude Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity, Volume 25 (2012) no. 10, pp. 2843-2873 | Article | MR 2979973 | Zbl 1251.35142

[8] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in 3 , Ann. of Math. (2), Volume 167 (2008) no. 3, pp. 767-865 | Article | MR 2415387 | Zbl 1178.35345

[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 | Article | MR 2259616 | Zbl 1122.35133

[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 | Article | MR 2993751 | Zbl 1241.35003

[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 | Article | MR 2482120 | Zbl 1156.35087

[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 | MR 839728 | Zbl 0535.35069

[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 | Article | MR 2231117 | Zbl 1119.35084

[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 | Article | MR 2360438

[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 | Article | MR 2559713 | Zbl 1180.35481

[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 | Article | MR 1741610 | Zbl 1004.35112

[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 | Article | MR 2257393 | Zbl 1115.35125

[18] Strauss, Walter Nonlinear Scattering Theory at Low Energy, J. Func. Anal., Volume 41 (1981), pp. 110-133 | MR 614228 | Zbl 0466.47006

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