论文标题
广义Navier-Stokes-Alpha模型的渐近行为以及相关模型的应用
Asymptotic behavior of a generalized Navier-Stokes-alpha model and applications to related models
论文作者
论文摘要
我们在整个三维空间中考虑了广义的α型模型，并由固定的（时间无关）的外力驱动。该模型包含特定情况的流体动力学一些相关方程，其中包括Navier-Stokes-Bardina的模型，关键的Alpha模型，分数和经典的Navier-Stokes方程，并具有附加的拖动/摩擦项。首先，我们研究了有限能源解决方案的独特性，在某些情况下研究了存在。然后，我们使用一个通用框架来研究其长时间行为，相对于相位空间的弱拓扑和强大的拓扑。当知道解决方案的独特性时，我们证明了强大的全球吸引子的存在。此外，在解决方案的独特性是未知的情况下，我们证明了一个薄弱的全球吸引子的存在。 弱/全球吸引子包含了我们模型的一种特殊的解决方案，即所谓的固定解决方案。在所有通用性中，我们构建了这些解决方案，并在模型中的某些物理常数足够大的情况下研究它们的独特性，轨道和渐近稳定性。作为双产品，我们表明在某些情况下，弱/全球吸引子降低到独特的固定溶液。
We consider a generalized alpha-type model in the whole three-dimensional space and driven by a stationary (time-independent) external force. This model contains as particular cases some relevant equations of the fluid dynamics, among them the Navier-Stokes-Bardina's model, the critical alpha-model, the fractional and the classical Navier-Stokes equations with an additional drag/friction term. First, we study the existence and in some cases the uniqueness of finite energy solutions. Then, we use a general framework to study their long time behavior with respect to the weak and the strong topology of the phase space. When the uniqueness of solutions is known, we prove the existence of a strong global attractor. Moreover, we proof the existence of a weak global attractor in the case when the uniqueness of solutions is unknown. The weak/global attractor contains a particular kind of solutions to our model, so-called the stationary solutions. In all generality we construct these solutions, and we study their uniqueness, orbital and asymptotic stability in the case when some physical constants in our model are large enough. As a bi-product, we show that in some cases the weak/global attractor reduces down to the unique stationary solution.
