Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations

Constantin, Peter; Kukavica, Igor; Vicol, Vlad

doi:10.1016/j.anihpc.2015.07.002

Constantin, Peter; Kukavica, Igor; Vicol, Vlad

Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 6, pp. 1569-1588.

Abstract

We consider the incompressible Euler equations on $R^{d}$ or $T^{d}$ , where $d \in {2, 3}$ . We prove that:

(a) In Lagrangian coordinates the equations are locally well-posed in spaces with fixed real-analyticity radius (more generally, a fixed Gevrey-class radius).

(b) In Lagrangian coordinates the equations are locally well-posed in highly anisotropic spaces, e.g. Gevrey-class regularity in the label $a_{1}$ and Sobolev regularity in the labels $a_{2}, \dots, a_{d}$ .

MR Zbl

DOI: 10.1016/j.anihpc.2015.07.002

Classification: 35Q35, 35Q30, 76D09
Mots-clés : Euler equations, Lagrangian and Eulerian coordinates, Analyticity, Gevrey class

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     author = {Constantin, Peter and Kukavica, Igor and Vicol, Vlad},
     title = {Contrast between {Lagrangian} and {Eulerian} analytic regularity properties of {Euler} equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1569--1588},
     publisher = {Elsevier},
     volume = {33},
     number = {6},
     year = {2016},
     doi = {10.1016/j.anihpc.2015.07.002},
     mrnumber = {3569243},
     zbl = {1353.35233},
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     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2015.07.002/}
}

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Constantin, Peter; Kukavica, Igor; Vicol, Vlad. Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations. Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 6, pp. 1569-1588. doi : 10.1016/j.anihpc.2015.07.002. https://www.numdam.org/articles/10.1016/j.anihpc.2015.07.002/

References
Cited by

[1] Bardos, C.; Benachour, S.; Zerner, M. Analycité des solutions périodiques de l'équation d'Euler en deux dimensions, C. R. Acad. Sci. Paris Sér. A–B, Volume 282 (1976), pp. A995–A998 | MR | Zbl

[2] Bardos, C.; Titi, E.S. Loss of smoothness and energy conserving rough weak solutions for the 3d Euler equations, Discrete Contin. Dyn. Syst., Ser. S, Volume 3 (2010) no. 2, pp. 185–197 | MR | Zbl

[3] A.L. Cauchy, Sur l'état du fluide à une époque quelconque du mouvement, in: Mémoires extraits des recueils de l'Académie des sciences de l'Institut de France, Théorie de la propagation des ondes à la surface dun fluide pesant dune profondeur indéfinie, Sciences mathématiques et physiques, Tome I, Seconde Partie, 1827.

[4] Chemin, J.-Y. Régularité de la trajectoire des particules d'un fluide parfait incompressible remplissant l'espace, J. Math. Pures Appl. (9), Volume 71 (1992) no. 5, pp. 407–417 | MR | Zbl

[5] Cheskidov, A.; Shvydkoy, R. Ill-posedness of basic equations of fluid dynamics in Besov spaces, Proc. Am. Math. Soc., Volume 138 (2010), pp. 1059–1067 | MR | Zbl

[6] Constantin, P. An Eulerian–Lagrangian approach for incompressible fluids: local theory, J. Am. Math. Soc., Volume 14 (2001) no. 2, pp. 263–278 (electronic) | DOI | MR | Zbl

[7] Constantin, P.; Vicol, V.; Wu, J. Analyticity of Lagrangian trajectories for well posed inviscid incompressible fluid models, 2014 | arXiv | MR

[8] DiPerna, R.J.; Majda, A.J. Oscillations and concentrations in weak solutions of the incompressible fluid equations, Commun. Math. Phys., Volume 108 (1987) no. 4, pp. 667–689 | DOI | MR | Zbl

[9] Euler, L. Principes généraux du mouvement des fluides, Académie Royale des Sciences et des Belles Lettres de Berlin, Mémoires, Volume 11 (1757), pp. 274–315

[10] Frisch, U.; Zheligovsky, V. A very smooth ride in a rough sea, Commun. Math. Phys., Volume 326 (2014) no. 2, pp. 499–505 | DOI | MR | Zbl

[11] Himonas, A.; Alexandrou, A.; Misiolek, G. Non-uniform dependence on initial data of solutions to the Euler equations of hydrodynamics, Commun. Math. Phys., Volume 296 (2010) no. 1, pp. 285–301 | DOI | MR | Zbl

[12] Gamblin, P. Système d'Euler incompressible et régularité microlocale analytique, Ann. Inst. Fourier (Grenoble), Volume 44 (1994) no. 5, pp. 1449–1475 | DOI | Numdam | MR | Zbl

[13] Glass, O.; Sueur, F.; Takahashi, T. Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid, Ann. Sci. Éc. Norm. Supér. (4), Volume 45 (2012) no. 1, pp. 1–51 | Numdam | MR | Zbl

[14] Kukavica, I.; Vicol, V. The domain of analyticity of solutions to the three-dimensional Euler equations in a half space, Discrete Contin. Dyn. Syst., Volume 29 (2011) no. 1, pp. 285–303 | DOI | MR | Zbl

[15] Kukavica, I.; Vicol, V. On the analyticity and Gevrey-class regularity up to the boundary for the Euler equations, Nonlinearity, Volume 24 (2011) no. 3, pp. 765–796 | DOI | MR | Zbl

[16] Misiolek, G.; Yoneda, T. Ill-posedness examples for the quasi-geostrophic and the Euler equations, Analysis, Geometry and Quantum Field Theory, Contemp. Math., vol. 584, Amer. Math. Soc., 2012, pp. 251–258 | DOI | MR | Zbl

[17] Misiolek, G.; Yoneda, T. Loss of continuity of the solution map for the Euler equations in α-modulation and Hl̈der spaces, 2014 | arXiv

[18] Nadirashvili, N. On stationary solutions of two-dimensional Euler equation, Arch. Ration. Mech. Anal., Volume 209 (2013) no. 3, pp. 729–745 | DOI | MR | Zbl

[19] Serfati, P. Structures holomorphes à faible régularité spatiale en mécanique des fluides, J. Math. Pures Appl., Volume 74 (1995) no. 2, pp. 95–104 | MR | Zbl

[20] Shnirelman, A. On the analyticity of particle trajectories in the ideal incompressible fluid, 2012 | arXiv | Zbl

[21] A. Shnirelman, Personal communication, 2015.

[22] Sueur, F. Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain, J. Differ. Equ., Volume 251 (2011) no. 12, pp. 3421–3449 | DOI | MR | Zbl

[23] Weber, H.M. Über eine Transformation der hydrodynamischen Gleichungen, J. Reine Angew. Math. (Crelle), Volume 68 (1868), pp. 286–292 | MR | Zbl

[24] Zheligovsky, V.; Frisch, U. Time-analyticity of Lagrangian particle trajectories in ideal fluid flow, J. Fluid Mech., Volume 749 (2014), pp. 404–430 | DOI | MR | Zbl

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