  In two spatial dimensions, there are very few global existence results for
the Kuramoto-Sivashinsky equation. The majority of the few results in the
literature are strongly anisotropic, i.e. are results of thin-domain type. In
the spatially periodic case, the dynamics of the Kuramoto-Sivashinsky equation
are in part governed by the size of the domain, as this determines how many
linearly growing Fourier modes are present. The strongly anisotropic results
allow linearly growing Fourier modes in only one of the spatial directions. We
provide here the first proof of global solutions for the two-dimensional
Kuramoto-Sivashinsky equation with a linearly growing mode in both spatial
directions. We develop a new method to this end, categorizing wavenumbers as
low (linearly growing modes), intermediate (linearly decaying modes which serve
as energy sinks for the low modes), and high (strongly linearly decaying
modes). The low and intermediate modes are controlled by means of a Lyapunov
function, while the high modes are controlled with operator estimates in
function spaces based on the Wiener algebra.
