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

Technical Summary: Single Exponential H1-Upper Bounds for the Primitive Equations

Problem Statement
The paper investigates the three-dimensional primitive equations governing large-scale geophysical flows (atmosphere and ocean) within a horizontally periodic box domain $\Omega = G \times (-h, 0)$. The system includes full viscosity and is subject to either homogeneous Neumann or Dirichlet boundary conditions on the upper and bottom boundaries, with periodic conditions on the lateral boundaries. The primary objective is to establish a priori bounds for strong solutions $v$ in the space $L^\infty(0, \infty; H^1(\Omega)) \cap L^2(0, \infty; \dot{H}^2(\Omega))$.

Specifically, the work addresses the growth rate of these bounds with respect to the initial data $a$ and time $t$. Existing literature, notably the global well-posedness results by Cao and Titi [3], provides upper bounds that grow double exponentially with respect to the initial data and time. Subsequent works [9, 10, 7] improved time dependence or addressed specific boundary conditions but generally retained at least double exponential dependence on the initial data or time, often relying on logarithmic Sobolev inequalities or specific smallness assumptions.

Methodology
The author employs a series of energy estimates and interpolation inequalities to derive the bounds, distinguishing the approach based on the boundary conditions:

1. Neumann Boundary Conditions (Proposition 1):

   • The proof utilizes $L^q$ estimates for the horizontal velocity $v$ (specifically $q=4, 8$) and $L^r$ estimates for the vertical derivative $v_z$ (specifically $r=2, 4$).
   • The method relies on the cancellation properties of the convection term and the Calderón–Zygmund inequality to handle the pressure term.
   • A key difficulty in this setting is the absence of the Poincaré inequality for $v$ (since the mean of $v$ is not necessarily zero). Consequently, the initial derivation yields a bound dependent on time ($e^{C\sqrt{t}}$).

2. Dirichlet Boundary Conditions (Theorem 1):

   • To achieve a time-independent bound, the author shifts the estimation strategy to $L^6$ estimates for $v$ and $L^3$ estimates for $v_z$.
   • A critical technical challenge in the Dirichlet case is the coupling between the pressure term $\pi$ and the boundary values. The pressure gradient $\nabla_H \pi$ is decomposed into $\nabla_H \pi_1$ (dependent on the bulk nonlinearity) and $\nabla_H \pi_2$ (dependent on the boundary trace of $v_z$).
   • The proof resolves the interdependence between the $L^q$ estimate for $v$ and the $L^r$ estimate for $v_z$ by carefully choosing the pair $(q, r) = (6, 3)$. This allows the boundary terms arising in one estimate to be controlled by the dissipation terms in the other.

3. Refinement for Neumann Case (Theorem 1):

   • To eliminate the time dependence found in Proposition 1, the author introduces the fluctuation variable $\tilde{v} = v - \bar{v}$ (where $\bar{v}$ is the vertical average).
   • Since $\tilde{v}$ satisfies a Poincaré-type inequality ($\|\tilde{v}\|_{L^q} \leq C \|v_z\|_{L^q}$), the analysis focuses on $L^6$ estimates for $\tilde{v}$ and $L^3$ estimates for $v_z$.
   • This approach effectively handles the "lower order terms" that previously caused time growth, leveraging the fact that the mean of the vertical derivative is zero.

Key Results
The paper establishes the following main theorem for both Neumann and Dirichlet boundary conditions:

For initial data $a \in H^1(\Omega)$ with $a_z \in L^3(\Omega)$ (and satisfying compatibility conditions), the strong solution $v$ satisfies:
$$\|v(t)\|_{H^1(\Omega)}^2 + \int_0^t \|\nabla^2 v(s)\|_{L^2(\Omega)}^2 \, ds \leq B(a) e^{C\|a\|_{L^2(\Omega)}^2}$$
for all $t \in [0, T]$, where $B(a)$ is a polynomial function of the norms of the initial data, and $C$ is a constant dependent only on the domain $\Omega$.

Crucially, the exponential factor depends only on the square of the $L^2$ norm of the initial data, $\|a\|_{L^2(\Omega)}^2$, and is independent of time $t$.

Significance and Claims
The paper claims two primary contributions:

1. Reduction of Exponential Growth: The derived upper bound is single exponential with respect to the initial data ($e^{C\|a\|_{L^2}^2}$), contrasting sharply with the double exponential bounds ($e^{e^{C\|a\|^2}}$) found in previous literature (e.g., [3, 9, 10, 7]). This represents a significant improvement in the quantitative understanding of the stability of the primitive equations.
2. Uniform-in-Time Estimates for Neumann Case: The paper provides what it claims to be the first uniform-in-time $H^1$ upper bound for the Neumann boundary condition case. In this setting, the standard Poincaré inequality is unavailable for the velocity field $v$, making the derivation of time-independent bounds non-trivial and previously unachieved in the literature.

The author notes that while the Dirichlet case allows for exponential decay as $t \to \infty$ if the Poincaré inequality is utilized, this specific decay behavior is not the primary focus of the current work. The results are presented as a priori estimates for strong solutions, assuming their existence, which is supported by prior local existence results.