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.

Classification: 35Q31, 76B03, 35B44, 76M60

Keywords: Euler equation, existence and regularity theory, blow-up, symbolic computation

@article{M2AN_2013__47_3_663_0, author = {Morosi, Carlo and Pernici, Mario and Pizzocchero, Livio}, title = {On power series solutions for the Euler equation, and the Behr-Ne\v cas-Wu initial datum}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {47}, number = {3}, year = {2013}, pages = {663-688}, doi = {10.1051/m2an/2012041}, mrnumber = {3056404}, language = {en}, url = {http://www.numdam.org/item/M2AN_2013__47_3_663_0} }

Morosi, Carlo; Pernici, Mario; 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, Volume 47 (2013) no. 3, pp. 663-688. doi : 10.1051/m2an/2012041. http://www.numdam.org/item/M2AN_2013__47_3_663_0/

[1] Padé approximants, 2nd edition, Cambridge University Press, Cambridge. Encycl. Math. Appl. 59 (1996). | Zbl 0923.41001

and ,[2] Sharp estimates for analytic pseudodifferential operators and application to Cauchy problems. J. Differ. Equ. 48 (1983) 241-268. | MR 696869 | Zbl 0535.35082

and ,[3] 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. | Numdam | MR 454413 | Zbl 0366.35022

and ,[4] Euler equations for incompressible ideal fluids. Russian Math. Surveys 62 (2007) 409-451. | MR 2355417 | Zbl 1139.76010

and ,[5] Remarks on the breakdown of smooth solutions for the 3D Euler equations. Commun. Math. Phys. 94 (1984) 61-66. | MR 763762 | Zbl 0573.76029

, and ,[6] On blow-up of solution for Euler equations. ESAIM: M2AN 35 (2001) 229-238. | Numdam | MR 1825697 | Zbl 0985.35057

, and ,[7] Éléments de Mathématique. Variétés différentielles et analytiques, Fascicule de résultats, Hermann, Paris (1971). | Zbl 1179.58001

,[8] Small scale structure of the Taylor-Green vortex. J. Fluid Mech. 130 (1983) 411-452. | Zbl 0517.76033

, , , , and ,[9] The Taylor-Green vortex and fully developed turbulence. J. Statist. Phys. 34 (1984) 1049-1063. | MR 751728

, , , , and ,[10] Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional ideal flows. Phys. Fluids A 4 (1992) 2845-2854. | Zbl 0775.76026

, , , and ,[11] 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). | MR 2954517 | Zbl 1273.35217

and ,[12] A posteriori regularity of the three-dimensional NavierStokes equations from numerical computations. J. Math. Phys. 48 (2007) 065-204. | MR 2337003 | Zbl 1144.81329

, , and ,[13] 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. | Zbl 0563.76057

,[14] Singularities of the Euler flow? Not out of the blue!. J. Stat. Phys. 113 (2003) 761-781. | MR 2036870 | Zbl 1058.76011

, and ,[15] 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. | MR 407477 | Zbl 0315.35077

,[16] Three-dimensional periodic flows with high-symmetry. J. Phys. Soc. Japan 54 (1985) 2132-2140.

,[17] Spontaneous singularity in three-dimensional inviscid, incompressible flow. Phys. Rev. Lett. 44 (1980) 572-574. | MR 558166

, and ,[18] Analytic functionals on the sphere. AMS, Providence. Transl. Math. Monogr. 178 (1998). | MR 1641900 | Zbl 0922.46040

,[19] On approximate solutions of semilinear evolution equations II. Generalizations, and applications to Navier-Stokes equations. Rev. Math. Phys. 20 (2008) 625-706. | MR 2433990 | Zbl 1162.35004

and ,[20] An H1 setting for the Navier-Stokes equations: Quantitative estimates. Nonlinear Anal. 74 (2011) 2398-2414. | MR 2781768 | Zbl 1209.35099

, ,[21] On approximate solutions of the incompressible Euler and Navier-Stokes equations. Nonlinear Anal. 75 (2012) 2209-2235. | MR 2870912 | Zbl 1236.35111

and ,[22] Extended series analysis of full octahedral flow: numerical evidence for hydrodynamic blowup. Fluid Dyn. Res. 33 (2003) 207-221. | MR 1995034 | Zbl 1032.76656

,[23] The convergence of diagonal Padé approximants and the Padé conjecture. J. Comput. Appl. Math. 86 (1997) 287-296. | MR 1491440 | Zbl 0888.41008

,[24] Padé approximants and efficient analytic continuation of a power series. Russian Math. Surveys 57 (2002) 43-141. | MR 1914542 | Zbl 1056.41005

,[25] Topological vector spaces, distributions and kernels. Academic Press, New York (1967). | MR 225131 | Zbl 0171.10402

,[26] 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.

,