On the time of existence of solutions of the Euler–Korteweg system

Audiard, Corentin

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 5, pp. 1139-1183.

Le système d’Euler–Korteweg est une perturbation dispersive du système d’Euler compressible classique. En dimension 3 et plus, sous une condition naturelle de stabilité de la pression, l’auteur a prouvé avec B. Haspot la nature globalement bien posée du problème pour des données initiales petites et irrotationnnelles. On continue ici ce travail en considérant le cas de données avec une petite composante rotationnelle, on prouve une borne inférieure sur le temps d’existence qui ne dépend que de cette composante. Dans la limite irrotationnelle on retrouve le résultat précédent d’existence globale.

Indépendamment, on construit dans un cas particulier des solutions devenant singulières en temps fini.

The Euler–Korteweg system is a dispersive perturbation of the usual compressible Euler equations. In dimension at least three, under a natural stability condition on the pressure, the author proved with B. Haspot that the Cauchy problem is globally well-posed for small, smooth, irrotational initial data. As a continuation of this work, we prove that if the initial velocity has a small rotational part, there exists a lower bound on the time of existence that depends only on some norm of this rotational part. In the zero vorticity limit we recover the previous global well-posedness result.

Independently of this analysis, we also provide (in a special case) a simple example of solution that blows up in finite time.

Mots clés : Euler equations, capillarity, time of existence, dispersion

[1] Alazard, Thomas; Delort, Jean-Marc Global solutions and asymptotic behavior for two dimensional gravity water waves, Ann. Sci. Éc. Norm. Supér., Volume 48 (2015) no. 5, pp. 1149-1238 | DOI | MR

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

[3] Audiard, Corentin Small energy traveling waves for the Euler–Korteweg system, Nonlinearity, Volume 30 (2017) no. 9, pp. 3362-3399 | DOI

[4] Audiard, Corentin; Haspot, Boris Global Well-Posedness of the Euler–Korteweg System for Small Irrotational Data, Commun. Math. Phys., Volume 351 (2017) no. 1, pp. 201-247 | DOI | MR

[5] Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer, 2011, xvi+523 pages | DOI | MR

[6] Benzoni-Gavage, Sylvie; Chiron, David Long wave asymptotics for the Euler–Korteweg system, Rev. Mat. Iberoam., Volume 34 (2018) no. 1, pp. 245-304 | DOI | MR

[7] Benzoni-Gavage, Sylvie; Danchin, Raphaël; Descombes, Stéphane Well-posedness of one-dimensional Korteweg models, Electron. J. Differ. Equ. (2006), 59, 35 pages | MR

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

[9] Benzoni-Gavage, Sylvie; Danchin, Raphaël; Descombes, Stéphane; Jamet, Didier Structure of Korteweg models and stability of diffuse interfaces, Interfaces Free Bound., Volume 7 (2005) no. 4, pp. 371-414 | DOI

[10] Bergh, Jöran; Löfström, Jörgen Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer, 1976, x+207 pages | MR

[11] Béthuel, Fabrice; Danchin, Raphaël; Smets, Didier On the linear wave regime of the Gross–Pitaevskii equation., J. Anal. Math., Volume 110 (2010), pp. 297-338 | DOI | Zbl

[12] Bresch, Didier; Desjardins, Benoît; Lin, Chi-Kun On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Commun. Partial Differ. Equations, Volume 28 (2003) no. 3-4, pp. 843-868 | DOI | MR

[13] Bresch, Didier; Gisclon, Marguerite; Lacroix-Violet, Ingrid On Navier–Stokes–Korteweg and Euler–Korteweg Systems: Application to Quantum Fluids Models (https://arxiv.org/abs/1703.09460)

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

[15] Deng, Yu; Ionescu, Alexandru D.; Pausader, Benoît; Pusateri, Fabio Global solutions of the gravity-capillary water-wave system in three dimensions, Acta Math., Volume 219 (2017) no. 2, pp. 213-402 | DOI | MR

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

[17] Germain, Pierre; Masmoudi, Nader; Shatah, Jalal Global solutions for the gravity water waves equation in dimension 3, Ann. Math., Volume 175 (2012) no. 2, pp. 691-754 | DOI | MR

[18] Giesselmann, Jan; Lattanzio, Corrado; Tzavaras, Athanasios E. Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics, Arch. Ration. Mech. Anal., Volume 223 (2017) no. 3, pp. 1427-1484 | DOI | MR

[19] Ginsberg, Daniel On the lifespan of three-dimensional gravity water waves with vorticity (https://arxiv.org/abs/1812.01583)

[20] Grillakis, Manoussos; Shatah, Jalal; Strauss, Walter Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., Volume 74 (1987) no. 1, pp. 160-197 | DOI | MR

[21] Guo, Yan; Pausader, Benoît Global smooth ion dynamics in the Euler–Poisson system, Commun. Math. Phys., Volume 303 (2011) no. 1, pp. 89-125 | DOI | MR

[22] 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 | MR

[23] 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 | MR

[24] Ionescu, Alexandru D.; Lie, Victor Long term regularity of the one-fluid Euler–Maxwell system in 3D with vorticity, Adv. Math., Volume 325 (2018), pp. 719-769 | DOI | MR

[25] Klainerman, Sergiu; Ponce, Gustavo Global, small amplitude solutions to nonlinear evolution equations, Commun. Pure Appl. Math., Volume 36 (1983) no. 1, pp. 133-141 | DOI | MR

[26] Meyer, Yves; Coifman, Ronald R. Ondelettes et opérateurs. III Opérateurs multilinéaires, Actualités Mathématiques, Hermann, 1991, p. i-xii and 383–538 | MR