AbstractConsider the primitive equations on $$\mathbb {R}^2\times (z_0,z_1)$$
R
2
×
(
z
0
,
z
1
)
with initial data a of the form $$a=a_1+a_2$$
a
=
a
1
+
a
2
, where $$a_1 \in \mathrm{BUC}_\sigma (\mathbb {R}^2;L^1(z_0,z_1))$$
a
1
∈
BUC
σ
(
R
2
;
L
1
(
z
0
,
z
1
)
)
and $$a_2 \in L^\infty _\sigma (\mathbb {R}^2;L^1(z_0,z_1))$$
a
2
∈
L
σ
∞
(
R
2
;
L
1
(
z
0
,
z
1
)
)
. These spaces are scaling-invariant and represent the anisotropic character of these equations. It is shown that for $$a_1$$
a
1
arbitrary large and $$a_2$$
a
2
sufficiently small, this set of equations admits a unique strong solution which extends to a global one and is thus strongly globally well posed for these data provided a is periodic in the horizontal variables. The approach presented depends crucially on mapping properties of the hydrostatic Stokes semigroup in the $$L^\infty (L^1)$$
L
∞
(
L
1
)
-setting. It can be seen as the counterpart of the classical iteration schemes for the Navier–Stokes equations, now for the primitive equations in the $$L^\infty (L^1)$$
L
∞
(
L
1
)
-setting.
The primitive equations in the scaling-invariant space $$L^{\infty }(L^1)$$
