  Planet-forming disks are fundamental objects thought to be inherited from
large scale rotation, through the conservation of angular momentum during the
collapse of a prestellar dense core. We investigate the possibility for a
protostellar disk to be formed from a motionless dense core which contains
non-axisymmetric density fluctuations. The rotation is thus generated locally
by the asymmetry of the collapse. We study the evolution of the angular
momentum in a non-axisymmetric collapse of a dense core from an analytical
point of view. To test the theory, we perform three-dimensional simulations of
a collapsing prestellar dense core using adaptative mesh refinement. We start
from a non-axisymmetrical situation, considering a dense core with random
density perturbations that follow a turbulence spectrum. We analyse the
emerging disk comparing the angular momentum it contains with the one expected
from our analytic development. We study the velocity gradients at different
scales in the simulation as it is done with observations. We show that the
angular momentum in the frame of a stellar object which is not located at the
center of mass of the core is not conserved, due to inertial forces. Our
simulations of such non-axisymmetrical collapse quickly produce accretion disks
at the small scales in the core. The analysis of the kinematics at different
scales in the simulated core reveals projected velocity gradients of amplitudes
similar to the ones observed in protostellar cores, and which directions vary,
sometimes even reversing when small and large scales are compared. These
complex kinematics patterns appear in recent observations, and could be a
discriminating feature with models where rotation is inherited from large
scales. Our results from simulations without initial rotation are more
consistent with these recent observations than when solid-body rotation is
initially [abridged]
