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

TL;DR

This paper investigates power series solutions of the Euler equation on a 3D torus, revisiting previous computational results to analyze convergence and potential blow-up, ultimately suggesting solutions likely exist beyond earlier estimated blow-up times.

Contribution

The authors extend previous computer algebra calculations to 52 terms, analyze convergence radii, and challenge earlier interpretations of blow-up indications in the Euler equation solutions.

Findings

Convergence radius  is around 0.32 to 0.33, not necessarily indicating blow-up.

Solutions likely exist at least up to a time  > 0.47, beyond previous estimates.

Pade analysis provides weak evidence for blow-up at later times.

Abstract

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 in 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 \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 and 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 and 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) Pade' analysis gives a rather weak indication that the solution might blow up at a later time.