  Following Toomre & Kalnajs (1991), local models of slightly dissipative
self-gravitating disks show how inhomogeneous structures can be maintained over
several galaxy rotations. Their basic physical ingredients are self-gravity,
dissipation and differential rotation. In order to explore the structures
resulting from these processes on the kpc scale, local simulation of
self-gravitating disks are performed in this paper in 2D as well as in 3D. The
third dimension becomes a priori important as soon as matter clumping causes a
tight coupling of the 3D equations of motion. The physically simple and general
framework of the model permits to make conclusions beyond the here considered
scales. A time dependent affine coordinate system is used, allowing to
calculate the gravitational forces via a particle-mesh FFT-method, increasing
the performance with respect to previous direct force calculations. Persistent
patterns, formed by transient structures, whose intensity and morphological
characteristic depend on the dissipation rate are obtained and described. Some
of our simulations reveal first signs of mass-size and velocity dispersion-size
power-law relations, but a clear scale invariant behavior will require more
powerful computer techniques.
