## Vector potentials in three-dimensional non-smooth domains.(English)Zbl 0914.35094

Considering curl as an operator in $$H_0(\text{div}): =\{\Phi\in L_2\mid \text{div} \Phi=0\}$$, the $$L_2$$-orthogonal decomposition into the closure of the range of curl and its kernel space (the space of harmonic Dirichlet fields) becomes obvious. It is known that under rather weak assumptions a corresponding compact imbedding result holds which implies the closedness of the range and the finite-dimensionality of the kernel. Thus, in this case we have that elements $$\Phi\in H_0(\text{div})$$ can be written as $$\Phi=\text{curl} \Psi$$ with a so-called vector potential $$\Psi\in H_0(\text{div})$$ if and only if $$\Phi$$ is orthogonal to this finite-dimensional kernel.
In this paper, for the price of stricter regularity constraints on the boundary of the underlying 3-dimensional domain $$\Omega$$ the latter orthogonality condition is characterized in terms of boundary traces. As a second vector potential construction fields $$\Phi\in H_0(\text{div})$$ are considered, which are ‘tangential’ at the boundary. Here orthogonality to the corresponding kernel (the space of harmonic Neumann fields) is also characterized by trace conditions imposed on ‘cuts’ through the ‘handles’ of $$\Omega$$. For the latter case also a third vector potential construction (requiring the boundary to be $$C^{1,1})$$ is shown. The preference given to the trace characterization finds its proper justification in the discussion of the global regularity of the vector potential. This regularity is stated in terms of the Sobolev chain $$(H_s(\Omega))_{s\in\mathbb{R}}$$. In particular, the continuous imbedding into the Sobolev space $$H_1(\Omega)$$, i.e. the well-known Gaffney inequality, is recovered for $$C^{1,1}$$-boundaries.
The results are utilized in a concluding chapter to construct divergence-free finite-element approximations. As a particular application a variational formulation of the Stokes problem is discussed.
Reviewer: R.Picard (Dresden)

### MSC:

 35Q30 Navier-Stokes equations 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J20 Variational methods for second-order elliptic equations 76D07 Stokes and related (Oseen, etc.) flows
Full Text:

### References:

 [1] Sobolev Spaces, Academic Press, New York, 1975. · Zbl 0314.46030 [2] Agmon, Comm. Pure Appl. Math. 17 pp 35– (1964) · Zbl 0123.28706 [3] ’Potential representation of incompressible vector fields’, in: Nonlinear Problems in Applied Mathematics, SIAM, Philadelphia, 1995, pp. 43-49. [4] Bendali, J. Math. Anal. Appl. 107 pp 537– (1985) · Zbl 0591.35053 [5] and , ’Formulation du système de Stokes en potentiel vecteur’, Internal Report 79, C. M. A. P., Ecole Polytechnique, 1982. [6] ’Méthodes d’éléments finis mixtes pour les équations de Navier-Stokes’, Thèse de 3ème cycle, Université Pierre et Marie Curie, 1979. [7] Birman, Russ. Math. Surv. 42 pp 75– (1987) · Zbl 0653.35075 [8] Borchers, Hokkaido Math. J. 19 pp 67– (1990) · Zbl 0719.35014 [9] Costabel, Math. Meth. in the Appl. Sci. 12 pp 365– (1990) · Zbl 0699.35028 [10] Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics, Vol. 1341, Springer, Berlin, 1988. · Zbl 0668.35001 [11] Dauge, Integr. Equat. Oper. Theory 15 pp 227– (1992) · Zbl 0767.46026 [12] ’Étude des équations de la magnéto-hydrodynamique stationnaire et de leurapproximation par éléments finis’, Thèse de 3ème cycle, Université Pierre et Marie Curie, 1982. [13] ’Formulation en potentiel vecteur du système de Stokes dans un domaine de $$\mathbb{R}$$3’, Internal Report 83015, Laboratoire d’Analyse Numérique de l’Université Pierre et Marie Curie, 1983. [14] Dubois, SIAM J. Numer. Anal. 27 pp 1103– (1990) · Zbl 0717.65086 [15] and , Inégalités en mécanique et en physique, Dunod, Paris, 1972. [16] El Dabaghi, Numer. Math. 48 pp 561– (1986) · Zbl 0625.76009 [17] Foias, Ann. Sc. Norm. Sup. Pisa V pp 29– (1978) [18] Friedrichs, Comm. Pure Appl. Math. 8 pp 551– (1955) · Zbl 0066.07504 [19] ’Curl-conforming finite, element methods for Navier-Stokes equations with non-standard boundary conditions in $$\mathbb{R}$$3’, in: The Navier-Stokes Equations, Theory and Numerical Methods, Proc. Conf., Oberwolfach, 1988 (, and eds), Lecture Notes in Mathematics, Vol. 1431, Springer, 1990. [20] Girault, Math. Meth. in the Appl. Sci. 15 pp 345– (1992) · Zbl 0761.35083 [21] and , Finite Element Methods for the Navier-Stokes Equations, Theory and Algorithms, Springer, Berlin, 1986. · Zbl 0585.65077 [22] Gobert, J. Math. Anal. Appl. 36 pp 518– (1971) · Zbl 0221.35016 [23] Topologie des surfaces, PUF, Paris, 1971. · Zbl 0211.55901 [24] Elliptic Boundary Value Problems in Nonsmooth Domains, Pitman, London, 1985. [25] ’Compacité par compensation II’, in: Proc. Int. Meeting on Recent Methods in Non-Linear Analysis ( and , eds), Pitagora Editrice, Bologne, 1979, pp 245-256. [26] Les méthodes directes en théorie des équations elliptiques, Masson, Paris, 1967. [27] ’Mixed finite elements in $$\mathbb{R}$$3’, Internal Report 49, C. M. A. P., École Polytechnique, 1979. [28] Nedelec, Numer. Math. 39 pp 97– (1982) · Zbl 0488.76038 [29] Peetre, Ann. Inst. Fourier 16 pp 279– (1966) · Zbl 0151.17903 [30] Picard, Proc. Roy. Soc. Edinb. Sect. A 92 pp 165– (1982) · Zbl 0516.35023 [31] Saranen, Math. Scand. 51 pp 310– (1982) · Zbl 0524.35100 [32] Topics in Nonlinear Analysis, Publ. Math. d’Orsay, Université Paris-Sud, 1978. [33] Theory and Numerical Analysis of the Navier-Stokes Equations, North-Holland, Amsterdam, 1977. [34] Verfürth, Numer. Math. 50 pp 685– (1987) · Zbl 0596.76073 [35] Weber, Math. Meth. in the Appl. Sci. 2 pp 12– (1980) · Zbl 0432.35032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.