The Primitive Equations in the scaling invariant space $L^{\infty}(L^1)$
Yoshikazu Giga, Mathis Gries, Matthias Hieber, Amru Hussein, Takahito Kashiwabara
TL;DR
This work proves strong and global well-posedness for the primitive equations in the scaling-invariant space $L^\infty(R^2;L^1(J))$, treating rough anisotropic data. By developing $L^\infty(L^1)$-type estimates for the hydrostatic Stokes semigroup and its gradient, the authors construct a fixed-point scheme that yields a local mild solution for data $a=a_1+a_2$ with $a_1$ in $BUC_\sigma(R^2;L^1(J))$ and a small perturbation $a_2$ in $L^\infty_\sigma(R^2;L^1(J))$, and prove that horizontal periodicity upgrades this to a unique global strong solution. The approach draws a parallel with classical Navier–Stokes iteration schemes and highlights the role of anisotropic, scaling-invariant spaces in handling rough initial data. The results extend the well-posedness theory of the primitive equations to a broad class of data, providing a robust framework for understanding geophysical flows in critical spaces.
Abstract
Consider the primitive equations on $\R^2\times (z_0,z_1)$ with initial data $a$ of the form $a=a_1+a_2$, where $a_1 \in BUC_σ(\R^2;L^1(z_0,z_1))$ and $a_2 \in L^\infty_σ(\R^2;L^1(z_0,z_1))$ and where $BUC_σ(L^1)$ and $L^\infty_σ(L^1)$ denote the space of all solenoidal, bounded uniformly continuous and all solenoidal, bounded functions on $\R^2$, respectively, which take values in $L^1(z_0,z_1)$. These spaces are scaling invariant and represent the anisotropic character of these equations. It is shown that, if $\|a_2\|_{L^\infty_σ(L^1)}$ is sufficiently small, then this set of equations has a unique, local, mild solution. If in addition $a$ is periodic in the horizontal variables, then this solution is a strong one and extends to a unique, global, strong solution. The primitive equations are thus strongly and globally well-posed for these data. The approach depends crucially on mapping properties of the hydrostatic Stokes semigroup in the $L^\infty(L^1)$-setting and can thus be seen as the counterpart of the classical iteration schemes for the Navier-Stokes equations for the situation of the primitive equations.