Abstract
The Euler–Korteweg equations are a modification of the Euler equations that take into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schrödinger type equation. Local well-posedness (in subcritical Sobolev spaces) was obtained by Benzoni–Danchin–Descombes in any space dimension, however, except in some special case (semi-linear with particular pressure) no global well-posedness is known. We prove here that under a natural stability condition on the pressure, global well-posedness holds in dimension \({d \geq 3}\) for small irrotational initial data. The proof is based on a modified energy estimate, standard dispersive properties if \({d \geq 5}\), and a careful study of the structure of quadratic nonlinearities in dimension 3 and 4, involving the method of space time resonances.
Résumé
Les équations d’Euler–Korteweg sont une modification des équations d’Euler prenant en compte l’effet de la capillarité. Dans le cas général elles forment un système quasi-linéaire qui peut se reformuler comme une équation de Schrödinger dégénérée. L’existence locale de solutions fortes a été obtenue par Benzoni–Danchin–Descombes en toute dimension, mais sauf cas très particuliers il n’existe pas de résultat d’existence globale. En dimension au moins 3, et sous une condition naturelle de stabilité sur la pression on prouve que pour toute donnée initiale irrotationnelle petite, la solution est globale. La preuve s’appuie sur une estimation d’énergie modifiée. En dimension au moins 5 les propriétés standard de dispersion suffisent pour conclure tandis que les dimensions 3 et 4 requièrent une étude précise de la structure des nonlinéarités quadratiques pour utiliser la méthode des résonances temps espaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Antonelli P., Marcati P.: On the finite energy weak solutions to a system in quantum fluid dynamics. Commun. Math. Phys. 287(2), 657–686 (2009)
Antonelli P., Marcati P.: The quantum hydrodynamics system in two space dimensions. Arch. Ration. Mech. Anal. 203(2), 499–527 (2012)
Audiard C.: Dispersive smoothing for the Euler–Korteweg model. SIAM J. Math. Anal. 44(4), 3018–3040 (2012)
Audiard, C., Haspot, B.: From Gross–Pitaevskii equation to Euler–Korteweg system, existence of global strong solutions with small irrotational initial data. Preprint. https://hal.archives-ouvertes.fr/hal-01077281
Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, Volume 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg (2011)
Benzoni-Gavage S., Danchin R., Descombes S.: On the well-posedness for the euler-korteweg model in several space dimensions. Indiana Univ. Math. J. 56, 1499–1579 (2007)
Benzoni-Gavage S., Danchin R., Descombes S., Jamet D.: Structure of Korteweg models and stability of diffuse interfaces. Interfaces Free Bound. 7(4), 371–414 (2005)
Benzoni-Gavage Sylvie: Planar traveling waves in capillary fluids. Differ. Integral Equ. 26(3–4), 439–485 (2013)
Bulíček M., Feireisl E., Málek J., Shvydkoy R.: On the motion of incompressible inhomogeneous Euler–Korteweg fluids. Discrete Contin. Dyn. Syst. Ser. S 3(3), 497–515 (2010)
Carles R., Danchin R., Saut J.-C.: Madelung, Gross–Pitaevskii and Korteweg. Nonlinearity 25(10), 2843–2873 (2012)
Cohn S.: Global existence for the nonresonant Schrödinger equation in two space dimensions. Can. Appl. Math. Q. 2(3), 257–282 (1994)
De Lellis C., Székelyhidi L. Jr.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1), 225–260 (2010)
Donatelli D., Feireisl E., Marcati P.: Well/ill posedness for the Euler–Korteweg–Poisson system and related problems. Commun. Partial Differ. Equ. 40(7), 1314–1335 (2015)
Germain P., Masmoudi N., Shatah J.: Global solutions for 2D quadratic Schrödinger equations. J. Math. Pures Appl. (9) 97(5), 505–543 (2012)
Germain P., Masmoudi N., Shatah J.: Global solutions for the gravity water waves equation in dimension 3. Ann. Math. (2) 175(2), 691–754 (2012)
Germain, P., Masmoudi, N., Shatah, J.: Global solutions for 3D quadratic Schrödinger equations. Int. Math. Res. Not. 2009(3), 414–432 (2009). doi:10.1093/imrn/rnn135
Germain P., Masmoudi N., Shatah J.: Global existence for capillary water waves. Commun. Pure Appl. Math. 68(4), 625–687 (2015)
Giesselman J., Lattanzio C., Tzavaras A.: Relative energy for the Korteweg theory and related hamiltonian flows in gas dynamics. Arch. Ration. Mech. Anal. 223, 1427 (2017)
Guo Y., Pausader B.: Global smooth ion dynamics in the Euler–Poisson system. Commun. Math. Phys. 303(1), 89–125 (2011)
Gustafson S., Nakanishi K., Tsai T.-P.: Scattering for the Gross–Pitaevskii equation. Math. Res. Lett. 13(2–3), 273–285 (2006)
Gustafson S., Nakanishi K., Tsai T.-P.: Global dispersive solutions for the Gross–Pitaevskii equation in two and three dimensions. Ann. Henri Poincaré 8(7), 1303–1331 (2007)
Gustafson S., Nakanishi K., Tsai T.-P.: Scattering theory for the Gross–Pitaevskii equation in three dimensions. Commun. Contemp. Math. 11(4), 657–707 (2009)
Hayashi, N., Naumkin, P.I.: On the quadratic nonlinear Schrödinger equation in three space dimensions. Int. Math. Res. Not. 2000(3), 115–132 (2000). doi:10.1155/S1073792800000088
Klainerman S., Ponce G.: Global, small amplitude solutions to nonlinear evolution equations. Commun. Pure Appl. Math. 36(1), 133–141 (1983)
Shatah J.: Global existence of small solutions to nonlinear evolution equations. J. Differ. Equ. 46(3), 409–425 (1982)
Shatah J.: Normal forms and quadratic nonlinear Klein–Gordon equations. Commun. Pure Appl. Math. 38(5), 685–696 (1985)
Strauss W.: Nonlinear scattering theory at low energy. J. Funct. Anal. 41, 110–133 (1981)
Taylor, M.E.: Partial Differential Equations III. Nonlinear Equations, Volume 117 of Applied Mathematical Sciences, 2nd edn. Springer, New York (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Mouhot
Rights and permissions
About this article
Cite this article
Audiard, C., Haspot, B. Global Well-Posedness of the Euler–Korteweg System for Small Irrotational Data. Commun. Math. Phys. 351, 201–247 (2017). https://doi.org/10.1007/s00220-017-2843-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-017-2843-8