Approximation of the Eigenvalues and of the Eigenfunctions for the Stokes Operator on an open Square

Bernd Rummler

We regard the Stokes eigenvalue problem (SEP) of the Stokes operator A on an open bounded square Q , where homogeneous Dirichlet boundary conditions are required. The square Q is a bounded, convex and C0,1-domain, and therefore one can use the regularity-results of [5] (resp. [4]) to specify 2 the regularity of the searched eigenfunctions w(x) by w(x) \in (W22(Q))2 \cap S1 with S1 := {u \in oW12(Q) : div u = 0 }. From this follows that one can suppose \phi \in oW22(Q) \cap W32(Q) for the stream functions (cf. [1] too) corresponding to the eigenfunctions w.

From the other hand, the Stokes eigenfunction are explicitly established only for two Coo-domains in R2 (cf. [6], [8]) at present, although the all-important features - like A is a selfadjoint opearator with an compact inverse - are true for A on Q too (cf. [2], [9]). According to [7], one knows that the Stokes eigenvalues satisfy \lambdaj \sim j |Q|-1 as j -> oo, but good approximations (upper and lower bounds) are available for the first eigenvalue \lambda1(Q) only (cf. [3]).

Our path to tackle the SEP on Q is the following: We formulate the problem in the framework of equivalent sequence spaces in the Fourier coefficients of the Fourier expansions of in half-periodic sinusoidal functions. So we get associated solenoidal vector functions w := (\phix2 , -\phix1 )T belonging to correspondent Sobolev spaces defined by weighted sequence spaces in the Fourier coefficients. We receive upper and lower bounds for all the \lambdaj(Q) as zeros of large determinants (fulfillment of boundary functionals) by Euler-Lagrange formulation of the SEP. Finally, we approximate the corresponding eigenfunctions wj by successive approximation. By way of example we show the approximated stream functions of w1 and w2.

References

  • Coffman, C. V., Duffin, R. J., "On the fundamental eigenfunctions of a clamped punctured disk", Adv.Appl.Math. 13(1992) 2 , 142-151 ;
  • Constantin, P., Foias, C., "Navier-Stokes Equations", Univ.of Chic.Press, Chicago (1988) ;
  • Goerisch, F., Haunhorst, H., "Eigenwertschranken f r Eigenwertaufgaben u mit partiellen Differentialgleichungen", ZAMM 65(1985) 3, 129-135 ;
  • Grisvard, P., "Elliptic Problems in Nonsmooth Domains", Boston: Pitman Publ. (1985);
  • Kellogg, R.B., Osborn, J.E., "A Regularity Result for the Stokes Problem in a Convex Polygon", J.Funct.Anal. 21, (1976), 397-431 ;
  • Lee, D. S.,Rummler, B., "The Eigenfunctions of the Stokes Operator in Special Domains III", ZAMM 82(2002) 6, 399-407 ;
  • Metivier, P.G., " Valueurs Propres D'Operateurs Definis par la Restriction de Systemes Variationnels a des Sous-Espaces", J.Math.pures et appl. 57, (1978), 133-156 ;
  • Rummler, B., "The Eigenfunctions of the Stokes Operator in Special Domains I", ZAMM 77(1997) 8, 619-627 ;
  • Temam, R., "Navier-Stokes equations" , Amsterdam: North Holland (1985)