no. 3
Partial differential equations
On the local existence for the Euler equations with free boundary for compressible and incompressible fluids
[Sur l'existence locale de solutions des équations d'Euler pour les fluides compressibles et incompressibles, avec frontière libre]

Présenté par : the Editorial Board

Disconzi, Marcelo M.1 ; Kukavica, Igor2
1 Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA
2 Dept. of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA
Comptes Rendus. Mathématique, Tome 356 (2018) no. 3, pp. 306-311.

Nous considérons les équations d'Euler compressibles et incompressibles avec frontière libre et tension de surface. Dans les deux cas, nous fournissons des estimations a priori pour l'existence de solutions locales avec vitesse initiale dans H3 et la condition H3 sur la densité dans le cas compressible. Une condition supplémentaire est nécessaire sur la frontière libre. Par comparaison avec la littérature, les deux résultats abaissent la régularité des données initiales pour les équations d'Euler en coordonnées lagrangiennes, avec tension de surface.

We consider the free boundary compressible and incompressible Euler equations with surface tension. In both cases, we provide a priori estimates for the local existence with the initial velocity in H3, with the H3 condition on the density in the compressible case. An additional condition is required on the free boundary. Compared to the existing literature, both results lower the regularity of initial data for the Lagrangian Euler equation with surface tension.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.02.002
Disconzi, Marcelo M. 1 ; Kukavica, Igor 2

1 Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA
2 Dept. of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA 
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Disconzi, Marcelo M.; Kukavica, Igor. On the local existence for the Euler equations with free boundary for compressible and incompressible fluids. Comptes Rendus. Mathématique, Tome 356 (2018) no. 3, pp. 306-311. doi : 10.1016/j.crma.2018.02.002. http://archive.numdam.org/articles/10.1016/j.crma.2018.02.002/

[1] Coutand, D.; Hole, J.; Shkoller, S. Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit, SIAM J. Math. Anal., Volume 45 (2013) no. 6, pp. 3690-3767

[2] Coutand, D.; Shkoller, S. Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., Volume 20 (2007) no. 3, pp. 829-930

[3] Disconzi, M.M.; Ebin, D.G. The free boundary Euler equations with large surface tension, J. Differ. Equ., Volume 261 (2016) no. 2, pp. 821-889

[4] M. Disconzi, I. Kukavica, A priori estimates for the free-boundary Euler equations with surface tension in three dimensions, preprint, 2017.

[5] M. Disconzi, I. Kukavica, A priori estimates for the 3d compressible free-boundary Euler equations with surface tension in the case of a liquid, preprint, 2017.

[6] Ignatova, M.; Kukavica, I. On the local existence of the free-surface Euler equation with surface tension, Asymptot. Anal., Volume 100 (2016), pp. 63-86

[7] Kukavica, I.; Tuffaha, A.; Vicol, V. On the local existence for the 3d Euler equation with a free interface, Appl. Math. Optim., Volume 76 (2017) no. 3, pp. 535-563 | DOI

[8] Schweizer, B. On the three-dimensional Euler equations with a free boundary subject to surface tension, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 22 (2005) no. 6, pp. 753-781

[9] Shatah, J.; Zeng, C. Geometry and a priori estimates for free boundary problems of the Euler equation, Commun. Pure Appl. Math., Volume 61 (2008) no. 5, pp. 698-744

[10] Shatah, J.; Zeng, C. Local well-posedness for fluid interface problems, Arch. Ration. Mech. Anal., Volume 199 (2011) no. 2, pp. 653-705

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