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Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations. (English) Zbl 1353.35233

Summary: We consider the incompressible Euler equations on \(\mathbb R^d\) or \(\mathbb T^d\), 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q30 Navier-Stokes equations
76D09 Viscous-inviscid interaction
35B65 Smoothness and regularity of solutions to PDEs
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