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 Ufimsk. Mat. Zh., 2011, Volume 3, Issue 1, Pages 53–79 (Mi ufa82)

Cauchy problem for the Navier–Stokes equations, Fourier method

R. S. Saks

Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa, Russia

Abstract: The Cauchy problem for the 3D Navier–Stokes equations with periodical conditions on the spatial variables is investigated. The vector functions under consideration are decomposed in Fourier series with respect to eigenfunctions of the curl operator. The problem is reduced to the Cauchy problem for Galerkin systems of ordinary differential equations with a simple structure. The program of reconstruction for these systems and numerical solutions of the Cauchy problems are realized. Several model problems are solved. The results are represented in a graphic form which illustrates the flows of the liquid. The linear homogeneous Cauchy problem is investigated in Gilbert spaces. Operator of this problem realizes isomorphism of these spaces. For a general case, some families of exact global solutions of the nonlinear Cauchy problem are found. Moreover, two Gilbert spaces with limited sequences of Galerkin approximations are written out.

Keywords: Fourier series, eigenfunctions of the curl operator, Navier–Stokes equations, Cauchy problem, global solutions, Galerkin systems, Gilbert spaces.

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English version:
Ufa Mathematical Journal, 2011, 3:1, 51–77 (PDF, 1468 kB)

Bibliographic databases:

Document Type: Article
UDC: 517.956.226

Citation: R. S. Saks, “Cauchy problem for the Navier–Stokes equations, Fourier method”, Ufimsk. Mat. Zh., 3:1 (2011), 53–79; Ufa Math. J., 3:1 (2011), 51–77

Citation in format AMSBIB
\Bibitem{Sak11} \by R.~S.~Saks \paper Cauchy problem for the Navier--Stokes equations, Fourier method \jour Ufimsk. Mat. Zh. \yr 2011 \vol 3 \issue 1 \pages 53--79 \mathnet{http://mi.mathnet.ru/ufa82} \zmath{https://zbmath.org/?q=an:1240.35389} \transl \jour Ufa Math. J. \yr 2011 \vol 3 \issue 1 \pages 51--77 

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This publication is cited in the following articles:
1. R. S. Saks, “Sobstvennye funktsii operatorov rotora, gradienta divergentsii i Stoksa. Prilozheniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 2(31) (2013), 131–146
2. Saks R.S., “Orthogonal Subspaces of the Space l-2(G) and Self-Adjoint Extensions of the Curl and Gradient-of-Divergence Operators”, Dokl. Math., 91:3 (2015), 313–317
3. Saks R.S., “The Gradient-of-Divergence Operator in l-2(G)”, Dokl. Math., 91:3 (2015), 359–363
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