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

• Published in: ESAIM. Mathematical modelling and numerical analysis Vol. 47; no. 3; p. 663
• Main Authors: Morosi, Carlo, Pernici, Mario, Pizzocchero, Livio
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.05.2013
• Subjects: Algebra; Cauchy problems; Eulers equations; Mathematical analysis; Studies; Vortices
• ISSN: 1290-3841
• DOI: 10.1051/m2an/2012041

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, NeÄas 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 [tau]3 in the H3 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 E. Behr, J. NeÄas 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 [tau]3, our results agree with the original computations of E. Behr, J. NeÄas and H. Wu, ESAIM: M2AN 35 (2001) 229-238 (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 Padé analysis, that the solution might blow up at a later time. [PUBLICATION ABSTRACT]