On power series solutions for the Euler equation, and the Behr-Necas-Wu initial datum

We consider the Euler equation for an incompressible fluid on a three dimensional torus, and the construction of its solution as a power series in time. We point out some general facts on this subject, from convergence issues for the power series to the role of symmetries of the initial datum. We then turn the attention to a paper by Behr, Necas and Wu (M2AN: Math. Model. Numer. Anal. 35 (2001), 229–238); here, the authors chose a very simple Fourier polynomial as an initial datum for the Euler equation and analyzed the power series in time for the solution, determining the first 35 terms by computer algebra. Their calculations suggested for the series a finite convergence radius tau_3 in the H^3 Sobolev space, with 0.32 < tau_3 < 0.35; they regarded this as an indication that the solution of the Euler equation blows up. We have repeated the calculations of Behr-Necas-Wu, using again computer algebra; the order has been increased from 35 to 52, using the symmetries of the initial datum to speed up computations. As for tau_3, our results agree with the original computations of Behr-Necas-Wu (yielding in fact to conjecture that 0.32 < tau_3 < 0.33). Moreover, our analysis supports the following conclusions: (a) The finiteness of tau_3 is not at all an indication of a possible blow-up. (b) There is a strong indication that the solution of the Euler equation does not blow up at a time close to tau_3. In fact, the solution is likely to exist, at least, up to a time theta_3 > 0.47. (c) There is a weak indication, based on Pade' analysis, that the solution might blow up at a later time.