  The primitive equations in a 3D infinite layer domain are considered with
linearly growing initial data in the horizontal direction, which illustrates
the global atmospheric rotating or straining flows. On the boundaries,
Dirichlet, Neumann or mixed boundary conditions are imposed. The
Ornstein-Uhlenbeck type operator appears in the linear parts, so the semigroup
theory is established by Trotter's arguments due to decomposition of
infinitesimal generators. To obtain smoothing properties of the semigroup,
derivatives of the associated kernel are calculated. For proving time-local
existence and uniqueness of mild solutions, the adapted Fujita-Kato scheme is
used in certain Sobolev spaces.
