AI 中文总结
研究二维平面上近静水平衡处仅具水平耗散的Boussinesq方程，利用速度-温度耦合产生内重力波及色散衰减与水平耗散构成稳定机制，并对特定初始扰动建立经典解全局存在唯一性及给出速度和温度衰减率。
AI 中文摘要
静水平衡是分层流体中的基本平衡态，在地球物理流体动力学中起核心作用。理解其在不完全耗散下的稳定性是长期挑战，因为仅各向异性扩散通常不足以控制非线性演化，且相应各向异性耗散的Navier-Stokes方程尚无强大稳定机制。本文研究二维平面上近静水平衡\((U,\Theta)=(0,x_2)\)处仅具水平耗散的Boussinesq方程。表明速度-温度耦合产生内重力波，其色散衰减与水平耗散共同提供有效稳定机制，弥补完全缺乏的垂直耗散。确定了色散波传播在不完全耗散流体系统中恢复稳定性的机制。对于\(k\geq14\)时\(H^k(\mathbb{R}^2)\cap W^{3,1}(\mathbb{R}^2)\)中足够小的初始扰动，建立了经典解的全局存在唯一性，以及速度和温度明确的各向异性、分量形式的长时间衰减率，包括垂直速度更快的衰减。
英文摘要
The hydrostatic balance is a fundamental equilibrium state in stratified fluids and plays a central role in geophysical fluid dynamics. Understanding its stability under incomplete dissipation is a longstanding challenge, since anisotropic diffusion alone is generally insufficient to control the nonlinear evolution and no robust stabilizing mechanism is known for the corresponding anisotropically dissipative Navier--Stokes equations. In this paper, we investigate the two-dimensional Boussinesq equations on $\mathbb{R}^2$ with only horizontal dissipation near the hydrostatic equilibrium $(U,Θ)=(0,x_2)$. We show that the velocity--temperature coupling generates internal gravity waves whose dispersive decay, together with the horizontal dissipation, provides an effective stabilizing mechanism that compensates for the complete absence of vertical dissipation. This identifies a mechanism by which dispersive wave propagation restores stability in an incompletely dissipative fluid system. For sufficiently small initial perturbations in $H^k(\mathbb{R}^2)\cap W^{3,1}(\mathbb{R}^2)$ with $k\ge14$, we establish the global existence and uniqueness of classical solutions together with explicit anisotropic, componentwise large-time decay rates for the velocity and temperature, including faster decay of the vertical velocity.
Comments40 pages, comments are welcome!
