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

Carlo Morosi; Mario Pernici; Livio Pizzocchero

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

## Access Full Article

top## Abstract

top## How to cite

topMorosi, 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 < 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.},

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 < 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.

LA - eng

KW - Euler equation; existence and regularity theory; blow-up; symbolic computation

UR - http://eudml.org/doc/273290

ER -

## References

top- [1] G.A. Baker and P. Graves-Morris, Padé approximants, 2nd edition, Cambridge University Press, Cambridge. Encycl. Math. Appl. 59 (1996). Zbl0923.41001
- [2] M.S. Baouendi and C. Goulaouic, Sharp estimates for analytic pseudodifferential operators and application to Cauchy problems. J. Differ. Equ.48 (1983) 241–268. Zbl0535.35082MR696869
- [3] C. Bardos and S. Benachour, Domaine d’analycité des solutions de l’équation d’Euler dans un ouvert de Rn. Annal. Scuola Norm. Sup. Pisa Cl. Sci.4 (1977) 647–687. Zbl0366.35022MR454413
- [4] C. Bardos and E.S. Titi, Euler equations for incompressible ideal fluids. Russian Math. Surveys62 (2007) 409–451. Zbl1139.76010MR2355417
- [5] J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3D Euler equations. Commun. Math. Phys.94 (1984) 61–66. Zbl0573.76029MR763762
- [6] E. Behr, J. Nečas and H. Wu, On blow-up of solution for Euler equations. ESAIM: M2AN 35 (2001) 229–238. Zbl0985.35057MR1825697
- [7] N. Bourbaki, Éléments de Mathématique. Variétés différentielles et analytiques, Fascicule de résultats, Hermann, Paris (1971). Zbl1179.58001
- [8] M.E. Brachet, D. Meiron, S. Orszag, B. Nickel, R. Morf and U. Frisch, Small scale structure of the Taylor–Green vortex. J. Fluid Mech.130 (1983) 411–452. Zbl0517.76033
- [9] M.E. Brachet, D. Meiron, S. Orszag, B. Nickel, R. Morf and U. Frisch, The Taylor–Green vortex and fully developed turbulence. J. Statist. Phys.34 (1984) 1049-1063. MR751728
- [10] M.E. Brachet, M. Meneguzzi, A. Vincent, H. Politano and P.L. Sulem, Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional ideal flows. Phys. Fluids A4 (1992) 2845–2854. Zbl0775.76026
- [11] T. Chen and N. Pavlović, A lower bound on blowup rates for the 3D incompressible Euler equation and a single exponential Beale–Kato–Majda estimate completer. ArXiv:1107.0435v1 [math.AP] (2011). Zbl1273.35217MR2954517
- [12] S.I. Chernyshenko, P. Constantin, J.C. Robinson and E.S. Titi, A posteriori regularity of the three-dimensional NavierStokes equations from numerical computations. J. Math. Phys.48 (2007) 065–204. Zbl1144.81329MR2337003
- [13] U. Frisch, Fully developed turbulence and singularities, in Chaotic Behavior of Deterministic Systems, edited by G. Iooss, R.H.G. Helleman, R. Stora. LesHouches, session XXXVI, North-Holland, Amsterdam (1983) 665–704. Zbl0563.76057
- [14] U. Frisch, T. Matsumoto and J. Bec, Singularities of the Euler flow? Not out of the blue!. J. Stat. Phys. 113 (2003) 761–781. Zbl1058.76011MR2036870
- [15] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral theory and differential equations, Proceedings of the Dundee Symposium. Lect. Notes Math. 448 (1975) 23-70. Zbl0315.35077MR407477
- [16] S. Kida, Three-dimensional periodic flows with high-symmetry. J. Phys. Soc. Japan54 (1985) 2132–2140.
- [17] R.H. Morf, S.A. Orszag and U. Frisch, Spontaneous singularity in three-dimensional inviscid, incompressible flow. Phys. Rev. Lett.44 (1980) 572-574. MR558166
- [18] M. Morimoto, Analytic functionals on the sphere. AMS, Providence. Transl. Math. Monogr. 178 (1998). Zbl0922.46040MR1641900
- [19] C. Morosi and L. Pizzocchero, On approximate solutions of semilinear evolution equations II. Generalizations, and applications to Navier–Stokes equations. Rev. Math. Phys. 20 (2008) 625–706. Zbl1162.35004MR2433990
- [20] C. Morosi, L. Pizzocchero, An H1 setting for the Navier–Stokes equations: Quantitative estimates. Nonlinear Anal.74 (2011) 2398–2414. Zbl1209.35099MR2781768
- [21] C. Morosi and L. Pizzocchero, On approximate solutions of the incompressible Euler and Navier–Stokes equations. Nonlinear Anal.75 (2012) 2209–2235. Zbl1236.35111MR2870912
- [22] R.B. Pelz, Extended series analysis of full octahedral flow: numerical evidence for hydrodynamic blowup. Fluid Dyn. Res.33 (2003) 207–221. Zbl1032.76656MR1995034
- [23] H. Stahl, The convergence of diagonal Padé approximants and the Padé conjecture. J. Comput. Appl. Math.86 (1997) 287–296. Zbl0888.41008MR1491440
- [24] S.P. Suetin, Padé approximants and efficient analytic continuation of a power series. Russian Math. Surveys57 (2002) 43–141. Zbl1056.41005MR1914542
- [25] F. Treves, Topological vector spaces, distributions and kernels. Academic Press, New York (1967). Zbl0171.10402MR225131
- [26] GMPY Collaboration, Multiprecision arithmetic for Python, http://code.google.com/p/gmpy. This software is a wrapper for GMP Multiple Precision Arithmetic Library, see http://gmplib.org.