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

Abstract

The three dimensional primitive equations with full viscosity are considered in a horizontally periodic box $\Omega$, 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 $L^{\infty }(0,\infty ;H^1(\Omega ))\cap L^2(0,\infty ;{\dot{H}}^2(\Omega ))$, the exponential part of which is $\exp (C\Vert a{\Vert }_{L^2(\Omega )}^2)$. 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.

10 pages