The Euler Equations in Planar Domains with Corners
The Euler Equations in Planar Domains with Corners
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DOI：
10.1007/s00205-019-01384-7
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发表时间：
2018-11
影响因子：
2.5
通讯作者：
C. Lacave;Andrej Zlatoš
中科院分区：
文献类型：
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作者：
C. Lacave;Andrej Zlatoš
当速度场不是已知全局几乎Lipschitz的先验时，只在某些特殊情况下建立了二维Euler方程解的整体唯一性，并且这些结果所适用的解具有涡度的扩散部分在速度不够规则的点附近是恒定的性质。假设后者最初成立，那么挑战是通过对拉格朗日轨迹的适当控制来沿着欧拉动力学传播这一性质。在具有钝角且在其他地方足够光滑的区域中，Yudovich解不能仅在这些角附近几乎是Lipschitz解，并且我们研究了涡度在那里保持不变的充要条件。我们证明，如果涡度最初在整个边界附近是恒定的，那么它将永远保持恒定(并且整体弱解是唯一的)，只要没有拐角大于。我们还表明，对于确实有这种角落的域，这通常是失败的。
When the velocity field is not a priori known to be globally almost Lipschitz, global uniqueness of solutions to the two-dimensional Euler equations has been established only in some special cases, and the solutions to which these results apply share the property that the diffuse part of the vorticity is constant near the points where the velocity is insufficiently regular. Assuming that the latter holds initially, the challenge is then to propagate this property along the Euler dynamic via an appropriate control of the Lagrangian trajectories. In domains with obtuse corners and sufficiently smooth elsewhere, Yudovich solutions fail to be almost Lipschitz only near these corners, and we investigate the necessary and sufficient conditions for the vorticity to remain constant there. We show that if the vorticity is initially constant near the whole boundary, then it remains so forever (and global weak solutions are unique), provided that no corner has angle greater than. We also show that this fails in general for domains that do have such corners.