Article
Ufa Mathematical Journal
Volume 3, Number 1, pp. 51-77
Cauchy problem for the Navier-Stokes equations, Fourier method.
Saks R.S.
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The Cauchy problem for the 3D Navier-Stokes equations with condition of periodicity on spatial variable is studied. The given and required vector functions are decomposed in Fourier series on eigenfunctions of the curl operator. The problem is reduced to a Cauchy problem for systems of the ordinary differential equations, which has a simple structure. The program for reconstruction of the Galerkin systems are made and program for the numerical solution of its Cauchy problem. Some modeling problems are calculated. The results are made out as the diagrams giving representation about movement of these flows of a liquid.
The linear homogeneous Cauchy problem is investigated in Gilbert spaces. It's proved operator of the problem realizes isomorphism of these spaces.
For general nonlinear Cauchy problem families of the exact global solutions are written out and also Gilbert spaces, in which the Galerkin approximations are limited.