规律性的“大”三维鲑鱼的行星地转模型的海洋动力学

C. Cao, E. Titi
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引用次数: 5

摘要

众所周知,具有垂直流体静力平衡和水平瑞利摩擦/阻尼、热扩散和热输运耦合的三维无粘性行星地转模型在数学上是病态的。这是因为非正流式物理边界条件隐式地产生了侧向边界温度的附加边界条件。由于科里奥利强迫项的存在,这一附加边界条件与无热通量物理边界条件不同。因此,二阶抛物热方程在两种不同的边界条件下是过定的。在以前的工作中,我们提出了一种补救方法,即在热传递方程中引入四阶人工超扩散,并证明了所提出模型的全局正则性。这种高阶扩散的一个缺点是热方程的最大/最小原理的损失。对于这个问题,R. Salmon提出了另一种解决办法,他在流体静力平衡方程中引入了一个额外的瑞利式摩擦/阻尼项来表示速度的垂直分量。本文证明了三维Salmon海洋动力学行星地转模型的强解在所有时间和所有初始数据下的全局性和适定性。也就是说,我们证明了该模型的强解对初始数据的全局存在唯一性和连续依赖性。与三维粘性PG模型不同,我们仍然无法证明弱解的唯一性。值得注意的是,我们还证明了萨尔蒙提出的附加阻尼项在何种意义上消除了原系统中的不适定性;因此,它可以被看作是“正则化”的术语,可以用来正则化其他相关系统。
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Regularity “in Large” for the 3D Salmon’s Planetary Geostrophic Model of Ocean Dynamics
Abstract It is well known, by now, that the three-dimensional non-viscous planetary geostrophic model, with vertical hydrostatic balance and horizontal Rayleigh friction/damping, coupled to the heat diffusion and transport, is mathematically ill-posed. This is because the no-normal flow physical boundary condition implicitly produces an additional boundary condition for the temperature at the lateral boundary. This additional boundary condition is different, because of the Coriolis forcing term, than the no-heat-flux physical boundary condition. Consequently, the second order parabolic heat equation is over-determined with two different boundary conditions. In a previous work we proposed one remedy to this problem by introducing a fourth-order artificial hyper-diffusion to the heat transport equation and proved global regularity for the proposed model. A shortcoming of this higher-oder diffusion is the loss of the maximum/minimum principle for the heat equation. Another remedy for this problem was suggested by R. Salmon by introducing an additional Rayleigh-like friction/damping term for the vertical component of the velocity in the hydrostatic balance equation. In this paper we prove the global, for all time and all initial data, well-posedness of strong solutions to the three-dimensional Salmon’s planetary geostrophic model of ocean dynamics. That is, we show global existence, uniqueness and continuous dependence of the strong solutions on initial data for this model. Unlike the 3D viscous PG model, we are still unable to show the uniqueness of the weak solution. Notably, we also demonstrate in what sense the additional damping term, suggested by Salmon, annihilate the ill-posedness in the original system; consequently, it can be viewed as “regularizing” term that can possibly be used to regularize other related systems.
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