Volume 47, Number 3, May-June 2013
|Page(s)||663 - 688|
|Published online||04 March 2013|
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
2 Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy
3 Dipartimento di Matematica, Università di Milano, Via C. Saldini 50, 20133 Milano, Italy
4 Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy
Revised: 17 July 2012
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.
Mathematics Subject Classification: 35Q31 / 76B03 / 35B44 / 76M60
Key words: Euler equation / existence and regularity theory / blow-up / symbolic computation
© EDP Sciences, SMAI, 2013
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.