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首页> 外文期刊>SIAM Journal on Mathematical Analysis >Stability of two-dimensional viscous incompressible flows under three-dimensional perturbations and inviscid symmetry breaking
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Stability of two-dimensional viscous incompressible flows under three-dimensional perturbations and inviscid symmetry breaking

机译：三维扰动和无粘性对称破坏下的二维粘性不可压缩流的稳定性

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In this article we consider weak solutions of the three-dimensional incompressible fluid flow equations with initial data admitting a one-dimensional symmetry group. We examine both the viscous and inviscid cases. For the case of viscous flows, we prove that Leray-Hopf weak solutions of the three-dimensional Navier-Stokes equations preserve initially imposed symmetry and that such symmetric flows are stable under general three-dimensional perturbations, globally in time. We work in three different contexts: two-and-a-half-dimensional, helical, and axisymmetric flows. In the inviscid case, we observe that as a consequence of recent work by De Lellis and Székelyhidi, there are genuinely three-dimensional weak solutions of the Euler equations with two-dimensional initial data. We also present two partial results where restrictions on the set of initial data and on the set of admissible solutions rule out spontaneous symmetry breaking; one is due to P.-L. Lions and the other is a consequence of our viscous stability result.
机译：在本文中，我们考虑了带有一维对称组的初始数据的三维不可压缩流体流动方程的弱解。我们同时检查了粘性和无粘性的情况。对于粘性流，我们证明了三维Navier-Stokes方程的Leray-Hopf弱解保留了最初施加的对称性，并且这种对称流在全局三维扰动下在全局上是稳定的。我们在三种不同的情况下工作：二维半流，螺旋流和轴对称流。在无粘性的情况下，我们观察到，由于De Lellis和Székelyhidi最近的工作，有二维初始数据的Euler方程确实存在三维弱解。我们还给出了两个部分结果，其中对初始数据集和可允许解集的限制排除了自发对称破坏。一个是由于P.-L。狮子和另一只是我们粘性稳定的结果。

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