Description: |
In his 1815 manuscript, Cauchy showed that the equations governing the motion of particles in ideal fluid flow can be once integrated in time. This yields what is now known as the Cauchy invariants. The invariants are then used to derive recurrence relations for the time-Taylor coefficients of the Lagrangian map from initial fluid particle positions to current ones. We use these relations for two purposes: On the one hand, we prove in an elementary way that Lagrangian solutions to the Euler equation for three-dimensional flows of ideal incompressible fluid have finite-time time-analyticity even for initial flows with limited spatial smoothness (for instance, for an initial vorticity of space-periodic flow that is just an absolutely summable Fourier series). We also show that the same time-analyticity result holds for compressible flows relevant in cosmology, which are governed by the Euler-Poisson equations. On the other, we discuss a numerical method for relatively fast integration of the Euler equation by summing up the time-Taylor series for the Lagrangian solution.
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