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

Carlo Morosi; Mario Pernici; Livio Pizzocchero

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 3, page 663-688
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Morosi, Carlo, Pernici, Mario, and Pizzocchero, Livio. "On power series solutions for the Euler equation, and the Behr–Nečas–Wu initial datum." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.3 (2013): 663-688. <http://eudml.org/doc/273290>.

@article{Morosi2013,
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 &lt; τ3 &lt; 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 &lt; τ3 &lt; 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 &gt; 0.47. (c) There is a weak indication, based on Padé analysis, that the solution might blow up at a later time.},
author = {Morosi, Carlo, Pernici, Mario, Pizzocchero, Livio},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Euler equation; existence and regularity theory; blow-up; symbolic computation},
language = {eng},
number = {3},
pages = {663-688},
publisher = {EDP-Sciences},
title = {On power series solutions for the Euler equation, and the Behr–Nečas–Wu initial datum},
url = {http://eudml.org/doc/273290},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Morosi, Carlo
AU - Pernici, Mario
AU - Pizzocchero, Livio
TI - On power series solutions for the Euler equation, and the Behr–Nečas–Wu initial datum
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 3
SP - 663
EP - 688
AB - 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 &lt; τ3 &lt; 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 &lt; τ3 &lt; 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 &gt; 0.47. (c) There is a weak indication, based on Padé analysis, that the solution might blow up at a later time.
LA - eng
KW - Euler equation; existence and regularity theory; blow-up; symbolic computation
UR - http://eudml.org/doc/273290
ER -

References

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