# 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
*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*with

^{2}_{2}(Q))^{2}\cap S^{1}*S*. From this follows that one can suppose

^{1}:= {u \in^{o}W^{1}_{2}(Q) : div u = 0 }*\phi \in*for the stream functions (cf. [1] too) corresponding to the eigenfunctions w.

^{o}W^{2}_{2}(Q) \cap W^{3}_{2}(Q)
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*only (cf. [3]).

_{1}(Q)
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*as zeros of large determinants (fulfillment of boundary functionals) by Euler-Lagrange formulation of the SEP. Finally, we approximate the corresponding eigenfunctions w

_{j}(Q)_{j}by successive approximation. By way of example we show the approximated stream functions of w

_{1}and w

_{2}.

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