Local well posedness of the Euler-Korteweg equations on \({{\mathbb{T}}^d} \). (English) Zbl 1477.35141
Summary: We consider the Euler-Korteweg system with space periodic boundary conditions \(x \in{\mathbb{T}}^d\). We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity assumptions on the initial data.
MSC:
| 35Q31 | Euler equations |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 76B45 | Capillarity (surface tension) for incompressible inviscid fluids |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
Euler-Korteveg equations; local well-posedness; energy estimates; para-differential calculusReferences:
| [1] | Alazard, T.; Burq, N.; Zuily, C., On the water-wave equations with surface tension, Duke Math. J., 158, 3, 413-499 (2011) · Zbl 1258.35043 · doi:10.1215/00127094-1345653 |
| [2] | Alazard, T.; Métivier, G., Paralinearization of the Dirichlet-Neumann operator and regularity of three-dimensional water waves, Commun. PDEs, 34, 12, 1632-1704 (2009) · Zbl 1207.35082 · doi:10.1080/03605300903296736 |
| [3] | Alinhac, S., Gérard, P.: Pseudo-differential operators and the Nash-Moser theorem. AMS, Graduate Studies in Mathematics, vol. 82. ISBN-10: 0-8218-3454-1 (2007) · Zbl 1121.47033 |
| [4] | 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) · Zbl 1177.82127 · doi:10.1007/s00220-008-0632-0 |
| [5] | Antonelli, P.; Marcati, P., The quantum hydrodynamics system in two space dimensions, Arch. Ration. Mech. Anal., 203, 499-527 (2012) · Zbl 1290.76165 · doi:10.1007/s00205-011-0454-7 |
| [6] | Antonelli, P., Marcati, P., Zheng, H.: genuine hydrodynamic analysis to the 1-D QHD system: existence, dispersion and stability. arXiv:1910.08104 (2019) · Zbl 1467.35253 |
| [7] | Audiard, C.; Haspot, B., Global well-posedness of the Euler-Korteweg system for small irrotational data, Commun. Math. Phys, 351, 1, 201-247 (2017) · Zbl 1369.35047 · doi:10.1007/s00220-017-2843-8 |
| [8] | 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) · Zbl 1125.76060 · doi:10.1512/iumj.2007.56.2974 |
| [9] | Berti, M., Bolle, P.: Quasi-periodic solutions of nonlinear wave equations on \({{\mathbb{T}}^d} \). In: European Research Monographs, European Research Monographs, 374 pages. ISBN 978-3-03719-211-5, 10.4171/211 (2020) · Zbl 1454.37003 |
| [10] | Berti, M., Maspero, A., Murgante, F.: Long time existence results for the Euler-Korteweg equations. In Preparation |
| [11] | Berti, M., Delort, J.-M.: Almost global solutions of capillary-gravity water waves equations on the circle. In: UMI Lecture Notes, x+268 pages, ISBN 978-3-319-99486-4 (2018) · Zbl 1409.35002 |
| [12] | Berti, M.; Feola, R.; Franzoi, L., Quadratic life span of periodic gravity-capillary water waves, Water Waves (2020) · Zbl 1502.35103 · doi:10.1007/s42286-020-00036-8 |
| [13] | Bona, JL; Smith, R., The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. R. Soc. Lond. Ser. A, 278, 1287, 555-601 (1975) · Zbl 0306.35027 · doi:10.1098/rsta.1975.0035 |
| [14] | Bresch, D.; Desjardins, B.; Lin, C., On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Commun. Partial Differ. Equ., 28, 3-4, 843-868 (2003) · Zbl 1106.76436 · doi:10.1081/PDE-120020499 |
| [15] | Carles, R.; Danchin, R.; Saut, J-C, Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity, 25, 10, 2843-2873 (2012) · Zbl 1251.35142 · doi:10.1088/0951-7715/25/10/2843 |
| [16] | Deng, Y.; Ionescu, A.; Pausader, B.; Pusateri, F., Global solutions of the gravity-capillary water-wave system in three dimensions, Acta Math., 219, 2, 213-402 (2017) · Zbl 1397.35190 · doi:10.4310/ACTA.2017.v219.n2.a1 |
| [17] | Dunn, J.; Serrin, J., On the thermomechanics of interstitial working, Arch. Ration. Mech. Anal., 88, 2, 95-133 (1985) · Zbl 0582.73004 · doi:10.1007/BF00250907 |
| [18] | Feola, R., Iandoli, F.: Local well-posedness for quasi-linear NLS with large Cauchy data on the circle. Ann. l’Institut H. Poincaré (C) Anal. Non linéaire 36(1), 119-164 (2018) · Zbl 1430.35207 |
| [19] | Feola, R., Iandoli, F.: Local well-posedness for the quasi-linear Hamiltonian Schrödinger equation on tori. arXiv:2003.04815 · Zbl 1430.35207 |
| [20] | Kato, T., The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58, 181-205 (1975) · Zbl 0343.35056 · doi:10.1007/BF00280740 |
| [21] | Maspero, A.; Robert, D., On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms, J. Funct. Anal., 273, 2, 721-781 (2017) · Zbl 1366.35153 · doi:10.1016/j.jfa.2017.02.029 |
| [22] | Métivier, G.: Para-differential calculus and applications to the Cauchy problem for nonlinear systems. Edizioni della Normale, Pisa, xii+140 pp. ISBN: 978-88-7642-329-1 (2008) · Zbl 1156.35002 |
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