From Gross-Pitaevskii equation to Euler Korteweg system, existence of global strong solutions with small irrotational initial data
Audiard, Corentin; Haspot, Boris (2017), From Gross-Pitaevskii equation to Euler Korteweg system, existence of global strong solutions with small irrotational initial data, Annali di Matematica Pura ed Applicata, p. 40. 10.1007/s10231-017-0702-z
TypeArticle accepté pour publication ou publié
External document linkhttps://hal.archives-ouvertes.fr/hal-01077281
Journal nameAnnali di Matematica Pura ed Applicata
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Laboratoire Jacques-Louis Lions [LJLL]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)In this paper we prove the global well-posedness for small data for the Euler Korteweg system in dimension N ≥ 3, also called compressible Euler system with quantum pressure. It is formally equivalent to the Gross-Pitaevskii equation through the Madelung transform. The main feature is that our solutions have no vacuum for all time. Our construction uses in a crucial way some deep results on the scattering of the Gross-Pitaevskii equation due to Gustafson, Nakanishi and Tsai in [28, 29, 30]. An important part of the paper is devoted to explain the main technical issues of the scattering in  and we give a detailed proof in order to make it more accessible. Bounds for long and short times are treated with special care so that the existence of solutions does not require smallness of the initial data in H s , s > N 2 . The optimality of our assumptions is also discussed.
Subjects / KeywordsFluid mechanics; Korteweg tensor; Global existence; Long-time dynamics; Scattering; Schrödinger type equations
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