# Bernd Rummler

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

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
*C*^{0,1}-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 (W*^{2}_{2}(Q))^{2} \cap S^{1}
with *S*^{1} := {u \in ^{o}W^{1}_{2}(Q) : div u = 0 }. From this
follows that one can suppose *\phi \in *^{o}W^{2}_{2}(Q) \cap
W^{3}_{2}(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 C^{oo}-domains in R^{2} (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 *\lambda*_{j} \sim
j · |Q|^{-1} as *j -> oo*, but good approximations (upper and
lower bounds) are available for the first eigenvalue *\lambda*_{1}(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 := (\phi*_{x2} , -\phi_{x1} )^{T}
belonging to correspondent Sobolev spaces defined by weighted sequence spaces in
the Fourier coefficients. We receive upper and lower bounds for all the
*\lambda*_{j}(Q) as zeros of large determinants (fulfillment
of boundary functionals) by Euler-Lagrange formulation of the SEP. Finally,
we approximate the corresponding eigenfunctions w_{j} by successive approximation.
By way of example we show the approximated stream functions of w_{1} and
w_{2}.

### References

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- 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)