Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations

Peter Constantin; Igor Kukavica; Vlad Vicol

doi:10.1016/j.anihpc.2015.07.002

Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations

Peter Constantin, Igor Kukavica, Vlad Vicol

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider the incompressible Euler equations on R^d or T^d, where d∈{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 a₁ and Sobolev regularity in the labels a₂,…,a_d. (c) In Eulerian coordinates both results (a) and (b) above are false.

Original language	English (US)
Pages (from-to)	1569-1588
Number of pages	20
Journal	Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume	33
Issue number	6
DOIs	https://doi.org/10.1016/j.anihpc.2015.07.002
State	Published - Nov 1 2016

Keywords

Analyticity
Euler equations
Gevrey class
Lagrangian and Eulerian coordinates

ASJC Scopus subject areas

Analysis
Mathematical Physics
Applied Mathematics

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10.1016/j.anihpc.2015.07.002

Cite this

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title = "Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations",

abstract = "We consider the incompressible Euler equations on Rd or Td, where d∈{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 a1 and Sobolev regularity in the labels a2,…,ad. (c) In Eulerian coordinates both results (a) and (b) above are false.",

keywords = "Analyticity, Euler equations, Gevrey class, Lagrangian and Eulerian coordinates",

author = "Peter Constantin and Igor Kukavica and Vlad Vicol",

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T1 - Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations

AU - Constantin, Peter

AU - Kukavica, Igor

AU - Vicol, Vlad

PY - 2016/11/1

Y1 - 2016/11/1

N2 - We consider the incompressible Euler equations on Rd or Td, where d∈{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 a1 and Sobolev regularity in the labels a2,…,ad. (c) In Eulerian coordinates both results (a) and (b) above are false.

AB - We consider the incompressible Euler equations on Rd or Td, where d∈{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 a1 and Sobolev regularity in the labels a2,…,ad. (c) In Eulerian coordinates both results (a) and (b) above are false.

KW - Analyticity

KW - Euler equations

KW - Gevrey class

KW - Lagrangian and Eulerian coordinates

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Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations

Abstract

Keywords

ASJC Scopus subject areas

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