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

TL;DR

This paper proves that solutions to the 3D primitive equations with full viscosity have single exponential bounds in H^1, improving upon previous double exponential bounds, and provides new uniform estimates for Neumann boundary conditions.

Contribution

It establishes the first single exponential H^1 upper bounds for strong solutions of the primitive equations with full viscosity, including new uniform estimates for Neumann boundary conditions.

Findings

Single exponential bounds in H^1 for strong solutions

Improved bounds from double exponential to single exponential

New uniform-in-time estimates for Neumann boundary conditions

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 $\exp(C \|a\|_{L^2(\Omega)}^2)$. 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 Poincar\'e inequality is unavailable for $v$, seems to be new.