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, Nečas and
Wu, ESAIM: M2AN 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
τ3 in the H3 Sobolev space, with
0.32 < τ3 < 0.35; they regarded this as an indication
that the solution of the Euler equation blows up. We have repeated the calculations of E.
Behr, J. Nečas and H. Wu, ESAIM: M2AN 35 (2001) 229–238,
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
τ3, our results agree with the original computations of E.
Behr, J. Nečas and H. Wu, ESAIM: M2AN 35 (2001) 229–238
(yielding in fact to conjecture that 0.32 < τ3 < 0.33).
Moreover, our analysis supports the following conclusions: (a) The finiteness of
τ3 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 τ3. In fact, the solution is likely to exist, at
least, up to a time θ3 > 0.47. (c) There is a weak
indication, based on Padé analysis, that the solution might blow up at a later time.