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A variational approach for the vector potential formulation of the Stokes and Navier-Stokes problems in three dimensional domains. (English) Zbl 0591.35053
Using the Georgescu’s results on Hodge-Kodaira decomposition [V. Georgescu, Ann. Mat. Pura Appl., IV. Ser. 122, 159-198 (1979; Zbl 0432.58026)] a vorticity-vector potential formulation of the Stokes problem in three-dimensional simply connected domain is obtained. The usual boundary condition of vector-potential are imposed, and the divergence-free condition results. In the last part, the previous considerations are extended to the nonlinear Navier-Stokes system.
Reviewer: G.Pasa

##### MSC:
 35Q30 Navier-Stokes equations 35D05 Existence of generalized solutions of PDE (MSC2000)
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##### References:
 [1] Bernardi, C, Thèse de 3ème cycle, (1979), Paris VI [2] Ciarlet, P; Raviart, P.A, Mixed finite element method for the biharmonic equation, (), 125-145 [3] Dominguez, J.M, Formulations en potentiel vecteur du système de Stokes dans un domaine de $$R$$^3, () [4] Duvaut, G; Lions, J.L, LES inéquations en mécanique et en physique, (1972), Dunod Paris · Zbl 0298.73001 [5] Foias, C; Temam, R, Remarques sur LES équations de Navier-Stokes stationnaires et LES phénomènes successifs de bifurcation, Ann. scuola norm. sup. Pisa cl. sci., 5, No. 1, 29-63, (1978) · Zbl 0384.35047 [6] Friedrichs, K.O, Differential forms on Riemannian manifolds, Comm. pure appl. math., 8, 551-590, (1955) · Zbl 0066.07504 [7] Gallic, S, Thèse de 3ème cycle, (1982), Paris VI [8] Georgescu, V, Some boundary value problem for differential forms on compact Riemannian manifolds, Ann. mat. pura appl., 122, 159-198, (1979) · Zbl 0432.58026 [9] Girault, V; Raviart, P.A, Finite element approximation of the Navier-Stokes equations, (1979), Springer-Verlag Berlin/Heidelberg/New-York · Zbl 0396.65070 [10] Glowinski, R; Pironneau, O, On a mixed finite element approximation of the Stokes problem. I. convergence of the approximation solutions, Numer. math., 33, 397-424, (1979) · Zbl 0423.65059 [11] Nedelec, J.C, Elements finis mixtes incompressibles pour l’équation de Stokes dans $$R$$^3, Numer. math., 39, 97-112, (1982) · Zbl 0488.76038 [12] Peetre, J, An other approach to elliptic boundary value problems, Comm. pure appl. math., 14, 711-731, (1961) · Zbl 0104.07303 [13] Richardson, S.M; Cornish, A.R.H, Solution of three dimensional incompressible flow problems, J. fluid mech., 82, 309-319, (1977), part 2 · Zbl 0367.76029 [14] Tartar, L, Non linear partial differential equations using compactness methods, () · Zbl 0401.35014
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