Global Well-posedness of the 3D Primitive Equations with Only Horizontal Viscosity and Diffusion
Global Well-posedness of the 3D Primitive Equations with Only Horizontal Viscosity and Diffusion
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发表时间：
2014-06
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通讯作者：
C. Cao;Jinkai Li;E. Titi
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作者：
C. Cao;Jinkai Li;E. Titi
本文考虑了在水平动量方程中只考虑水平涡动粘性，在温度方程中只考虑水平扩散的三维行星海洋和大气动力学原始方程的初边值问题。对于任意H^2 $初始数据，建立了强解的整体适定性.利用一个N维对数Sobolev嵌入不等式和经典Gronwall不等式的系统形式建立了全局正则性的先验H^2估计，其中Sobolev嵌入不等式将一阶导数的Lq范数限制为Lp范数的对数.
In this paper, we consider the initial-boundary value problem of the 3D primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well-posedness of strong solution is established for any $H^2$ initial data. An $N$-dimensional logarithmic Sobolev embedding inequality, which bounds the $L^\infty$ norm in terms of the $L^q$ norms up to a logarithm of the $L^p$-norm, for $p>N$, of the first order derivatives, and a system version of the classic Gronwall inequality are exploited to establish the required a priori $H^2$ estimates for the global regularity.