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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