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

Published online by Cambridge University Press:  04 March 2013

Carlo Morosi ,
Mario Pernici  and
Livio Pizzocchero
Carlo Morosi
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, P.za L. da Vinci 32, 20133 Milano, Italy. [email protected]
Mario Pernici
Affiliation:
Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy; [email protected]
Livio Pizzocchero
Affiliation:
Dipartimento di Matematica, Università di Milano, Via C. Saldini 50, 20133 Milano, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy; [email protected]
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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 generalfacts on this subject, from convergence issues for the power series to the role ofsymmetries of the initial datum. We then turn the attention to a paper by Behr, Nečas andWu, ESAIM: M2AN 35 (2001) 229–238; here, the authors chose avery simple Fourier polynomial as an initial datum for the Euler equation and analyzed thepower series in time for the solution, determining the first 35 terms by computer algebra.Their calculations suggested for the series a finite convergence radiusτ3 in the H3 Sobolev space, with0.32 < τ3 < 0.35; they regarded this as an indicationthat 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 thesymmetries of the initial datum to speed up computations. As forτ3, 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 ofτ3 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 atime close to τ3. In fact, the solution is likely to exist, atleast, up to a time θ3 > 0.47. (c) There is a weakindication, based on Padé analysis, that the solution might blow up at a later time.

Keywords

Euler equationexistence and regularity theoryblow-upsymbolic computation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 3 , May 2013 , pp. 663 - 688
Copyright
© EDP Sciences, SMAI, 2013

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References

Baker, G.A. and Graves-Morris, P., Padé approximants, 2nd edition, Cambridge University Press, Cambridge. Encycl. Math. Appl. 59 (1996). Google Scholar
Baouendi, M.S. and Goulaouic, C., Sharp estimates for analytic pseudodifferential operators and application to Cauchy problems. J. Differ. Equ. 48 (1983) 241–268. Google Scholar
Bardos, C. and Benachour, S., 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. Google Scholar
Bardos, C. and Titi, E.S., Euler equations for incompressible ideal fluids. Russian Math. Surveys 62 (2007) 409–451. Google Scholar
Beale, J. T., Kato, T. and Majda, A., Remarks on the breakdown of smooth solutions for the 3D Euler equations. Commun. Math. Phys. 94 (1984) 61–66. Google Scholar
Behr, E., Nečas, J. and Wu, H., On blow-up of solution for Euler equations. ESAIM: M2AN 35 (2001) 229–238. Google Scholar
N. Bourbaki, Éléments de Mathématique. Variétés différentielles et analytiques, Fascicule de résultats, Hermann, Paris (1971).
Brachet, M.E., Meiron, D., Orszag, S., Nickel, B., Morf, R. and Frisch, U., Small scale structure of the Taylor–Green vortex. J. Fluid Mech. 130 (1983) 411–452. Google Scholar
Brachet, M.E., Meiron, D., Orszag, S., Nickel, B., Morf, R. and Frisch, U., The Taylor–Green vortex and fully developed turbulence. J. Statist. Phys. 34 (1984) 1049-1063. Google Scholar
Brachet, M.E., Meneguzzi, M., Vincent, A., Politano, H. and Sulem, P.L., Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional ideal flows. Phys. Fluids A 4 (1992) 2845–2854. Google Scholar
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).
Chernyshenko, S.I., Constantin, P., Robinson, J.C. and Titi, E.S., A posteriori regularity of the three-dimensional NavierStokes equations from numerical computations. J. Math. Phys. 48 (2007) 065–204. Google Scholar
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.
Frisch, U., Matsumoto, T. and Bec, J., Singularities of the Euler flow? Not out of the blue!. J. Stat. Phys. 113 (2003) 761–781. Google Scholar
Kato, T., 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. Google Scholar
Kida, S., Three-dimensional periodic flows with high-symmetry. J. Phys. Soc. Japan 54 (1985) 2132–2140. Google Scholar
Morf, R.H., Orszag, S.A. and Frisch, U., Spontaneous singularity in three-dimensional inviscid, incompressible flow. Phys. Rev. Lett. 44 (1980) 572-574. Google Scholar
Morimoto, M., Analytic functionals on the sphere. AMS, Providence. Transl. Math. Monogr. 178 (1998). Google Scholar
Morosi, C. and Pizzocchero, L., On approximate solutions of semilinear evolution equations II. Generalizations, and applications to Navier–Stokes equations. Rev. Math. Phys. 20 (2008) 625–706. Google Scholar
Morosi, C., Pizzocchero, L., An H 1 setting for the Navier–Stokes equations: Quantitative estimates. Nonlinear Anal. 74 (2011) 2398–2414. Google Scholar
Morosi, C. and Pizzocchero, L., On approximate solutions of the incompressible Euler and Navier–Stokes equations. Nonlinear Anal. 75 (2012) 2209–2235. Google Scholar
Pelz, R.B., Extended series analysis of full octahedral flow: numerical evidence for hydrodynamic blowup. Fluid Dyn. Res. 33 (2003) 207–221. Google Scholar
Stahl, H., The convergence of diagonal Padé approximants and the Padé conjecture. J. Comput. Appl. Math. 86 (1997) 287–296. Google Scholar
Suetin, S.P., Padé approximants and efficient analytic continuation of a power series. Russian Math. Surveys 57 (2002) 43–141. Google Scholar
F. Treves, Topological vector spaces, distributions and kernels. Academic Press, New York (1967).
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.