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

Takahito Kashiwabara

TL;DR

This work proves that the 3D primitive equations with full viscosity in a horizontally periodic box admit a unified $H^1$-level a priori bound that is single-exponential in the initial data and uniform in time for both Neumann and Dirichlet boundary conditions. The authors develop a hierarchical framework of $L^q$ and $L^r$ estimates for the horizontal velocity $v$, its vertical derivative $v_z$, and the horizontal gradient $\nabla_H v$, carefully handling pressure terms via Calderón–Zygmund arguments and Gronwall-type inequalities. For Dirichlet boundaries, the analysis leverages $L^4$, $L^6$ bounds and a pressure decomposition to obtain the desired bound; for Neumann boundaries, a mean-zero decomposition $\tilde{v}$ and a transition to $L^6$/$L^3$ estimates remove time-growth factors and yield a uniform-in-time estimate. The results improve on prior double-exponential bounds and have implications for long-time behavior and global well-posedness of geophysical flows under both boundary conditions.

Abstract

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.

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

TL;DR

This work proves that the 3D primitive equations with full viscosity in a horizontally periodic box admit a unified -level a priori bound that is single-exponential in the initial data and uniform in time for both Neumann and Dirichlet boundary conditions. The authors develop a hierarchical framework of and estimates for the horizontal velocity , its vertical derivative , and the horizontal gradient , carefully handling pressure terms via Calderón–Zygmund arguments and Gronwall-type inequalities. For Dirichlet boundaries, the analysis leverages , bounds and a pressure decomposition to obtain the desired bound; for Neumann boundaries, a mean-zero decomposition and a transition to / estimates remove time-growth factors and yield a uniform-in-time estimate. The results improve on prior double-exponential bounds and have implications for long-time behavior and global well-posedness of geophysical flows under both boundary conditions.

Abstract

The three dimensional primitive equations with full viscosity are considered in a horizontally periodic box , which are subject to either the homogeneous Neumann or Dirichlet conditions on the upper and bottom parts of the boundary. For a strong solution with initial data , we establish \emph{a priori} bounds in , the exponential part of which is . 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 , seems to be new.
Paper Structure (13 sections, 2 theorems, 80 equations)

This paper contains 13 sections, 2 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Proof of Proposition \ref{['main prop']}
  3. Step 1: estimates for $v \in L^\infty_t L^q(\Omega)$
  4. Step 2: estimates for $v_z := \partial_z v \in L^\infty_t L^r(\Omega) \, (r = 2, 4)$
  5. Step 3: estimate for $\nabla_H v \in L^\infty_t L^2(\Omega)$
  6. Proof of Theorem \ref{['main thm']} for Dirichlet BC case
  7. Step 1: estimates for $v \in L^\infty_t L^4(\Omega)$ and $v_z \in L^\infty_t L^2(\Omega)$
  8. Step 2: estimates for $v \in L^\infty_t L^6(\Omega)$ and $v_z \in L^\infty_t L^3(\Omega)$
  9. Step 3: estimate for $\nabla_H v \in L^\infty_t L^2(\Omega)$
  10. Proof of Theorem \ref{['sec1: proof of main thm']} for Neumann BC case
  11. Step 1: estimate for $\tilde{v} \in L^\infty_t L^q(\Omega) \, (q \in (2, 6])$
  12. Step 2: estimates for $v_z \in L^\infty_t L^r(\Omega) \, (r = 2, 3)$
  13. Step 3: estimate for $\nabla_H v \in L^\infty_t L^2(\Omega)$

Key Result

Theorem 1

Suppose that the initial data satisfies $a \in H^1(\Omega)$ and $a_z: = \partial_z a \in L^3(\Omega)$, as well as the compatibility conditions: $\operatorname{div}_H \bar{a} = 0$, the horizontal periodicity, and $a = 0$ on $\Gamma_b$ in case of (eq: Dirichlet BC). Then the solution of (eq: PEs)--(eq where $B(a)$ is a polynomial function of $\|a\|_{H^1(\Omega)}$ and $\|a_z\|_{L^3(\Omega)}$, and $C$

Theorems & Definitions (3)

  • Theorem 1
  • Remark 1
  • Proposition 1