Single exponential $H^1$-upper bounds for the primitive equations

The three dimensional primitive equations with full viscosity are considered in a horizontally periodic box $Ω$, which are subject to either the homogeneous Neumann or Dirichlet conditions on the upper and bottom parts of the boundary. For a strong solution $v$ with initial data $a$, we establish \emph{a priori} bounds in $L^\infty(0, \infty; H^1(Ω)) \cap L^2(0, \infty; \dot H^2(Ω))$, the exponential part of which is $\exp(C \|a\|_{L^2(Ω)}^2)$. This is in contrast to the upper bounds reported in the existing literature that are double exponential. Furthermore, the uniform-in-time estimate for the Neumann condition case, in which the Poincaré inequality is unavailable for $v$, seems to be new.