  There is little analytical theory for the behavior of solutions of the
Kuramoto-Sivashinsky equation in two spatial dimensions over long times. We
study the case in which the spatial domain is a two-dimensional torus. In this
case, the linearized behavior depends on the size of the torus -- in
particular, for different sizes of the domain, there are different numbers of
linearly growing modes. We prove that small solutions exist for all time if
there are no linearly growing modes, proving also in this case that the radius
of analyticity of solutions grows linearly in time. In the general case (i.e.,
in the presence of a finite number of growing modes), we make estimates for how
the radius of analyticity of solutions changes in time.
