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, Tome 33 (2016) no. 6, pp. 1569-1588

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Résumé

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 $a_{2}, \dots, a_{d}$ .

(c) In Eulerian coordinates both results (a) and (b) above are false.

MR Zbl

DOI : 10.1016/j.anihpc.2015.07.002

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

@article{AIHPC_2016__33_6_1569_0,
     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},
     year = {2016},
     publisher = {Elsevier},
     volume = {33},
     number = {6},
     doi = {10.1016/j.anihpc.2015.07.002},
     mrnumber = {3569243},
     zbl = {1353.35233},
     language = {en},
     url = {http://geodesic.mathdoc.fr/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, Tome 33 (2016) no. 6, pp. 1569-1588. doi: 10.1016/j.anihpc.2015.07.002

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