Data di Pubblicazione:
2012
Citazione:
On power series solutions for the Euler
equation, and the Behr-Necas-Wu initial
datum / C. Morosi, M. Pernici, L. Pizzocchero. - [s.l] : arXiv, 2012 Mar.
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, 2001 [5]; 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 [5], 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 [5] (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) Pade' analysis gives a rather weak indication that the solution might blow
up at a later time.
Tipologia IRIS:
08 - Relazione interna o rapporto di ricerca
Keywords:
Euler equation, existence and regularity theory, blow-up, symbolic computation.
Elenco autori:
C. Morosi, M. Pernici, L. Pizzocchero
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