On power series solutions for the Euler equation, and the Behr–Nečas–Wu initial datum

¹ Dipartimento di Matematica, Politecnico di Milano, P.za L. da Vinci 32, 20133 Milano, Italy
carlo.morosi@polimi.it
² Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy
mario.pernici@mi.infn.it
³ Dipartimento di Matematica, Università di Milano, Via C. Saldini 50, 20133 Milano, Italy
⁴ Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy
livio.pizzocchero@unimi.it

Received: 30 March 2012
Revised: 17 July 2012

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, 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.