Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations

Abstract

We consider the incompressible Euler equations on ${\mathbb{R}}^d$, 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 well-posed in highly anisotropic spaces, e.g.~Gevrey-class regularity in the label $a_1$ and Sobolev regularity in the labels $a_2,...,a_d$. (c) In Eulerian coordinates both results (a) and (b) above are false.

22 pages

References in corpus (1)

• Time-analyticity of Lagrangian particle trajectories in ideal fluid flow

Cited by in corpus (4)

• Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces
• A constructive approach to regularity of Lagrangian trajectories for incompressible Euler flow in a bounded domain
• WKB analysis of the Logarithmic Nonlinear Schrodinger Equation in an analytic framework
• Asymptotic expansions for the Lagrangian trajectories from solutions of the Navier-Stokes equations