论文标题
一般平面域的Euler方程
Euler Equations on General Planar Domains
论文作者
论文摘要
我们在可能的奇异平面域的几何形状上获得了足够的条件，该几何形状保证了对它们上涡流有限且最初在边界附近最初恒定的Euler方程的任何弱解决方案的全局唯一性。这种情况仅比排除角度大于$π$的角的限制性略大，尤其是所有凸形域满足。我们方法中的主要成分表明，边界附近涡度的恒定性一直保留在一直以来，因为这些域上的Euler粒子轨迹，即使对于一般有限的溶液，也无法在有限的时间内达到边界。然后，我们使用它来表明，对于一般有限解决方案，这种可能的奇异域的边界无法创建涡度。我们还表明，通过构建任意接近满足的域，以及粒子轨迹可以在有限的时间内到达边界，我们的状况本质上是敏锐的。另外，当满足条件时，我们发现粒子轨迹对边界的最快方法的渐近速率有锐利的边界。
We obtain a general sufficient condition on the geometry of possibly singular planar domains that guarantees global uniqueness for any weak solution to the Euler equations on them whose vorticity is bounded and initially constant near the boundary. This condition is only slightly more restrictive than exclusion of corners with angles greater than $π$ and, in particular, is satisfied by all convex domains. The main ingredient in our approach is showing that constancy of the vorticity near the boundary is preserved for all time because Euler particle trajectories on these domains, even for general bounded solutions, cannot reach the boundary in finite time. We then use this to show that no vorticity can be created by the boundary of such possibly singular domains for general bounded solutions. We also show that our condition is essentially sharp in this sense by constructing domains that come arbitrarily close to satisfying it, and on which particle trajectories can reach the boundary in finite time. In addition, when the condition is satisfied, we find sharp bounds on the asymptotic rate of the fastest possible approach of particle trajectories to the boundary.
