论文标题
关于2D Euler方程和应用的规模不变解决方案的长期行为
On the long-time behavior of scale-invariant solutions to the 2d Euler equation and applications
论文作者
论文摘要
我们研究了满足离散对称性的2D Euler方程的规模不变解的长期行为。我们表明，在$ \ mathbb {s}^1 $放松身份的所有比例不变的解决方案都具有有限变化的状态，即在零件恒定的状态下，有限的跳跃。 所有连续的尺度不变溶液在无限时间内变得奇异并均匀。在$ \ mathbb {r}^2 $上，这对应于通用无限的螺旋和尖式形成。我们分析的主要工具是发现单调数量，该数量可以测量从原点移开的粒子数量。 这种单调性也适用于本地的2D Euler方程的解决方案，该方程为$ m $倍对称（$ M \ geq 4 $），并且在对称点具有径向限制。 我们的结果也适用于大型革命表面上的Euler方程（例如$ \ Mathbb {S}^2 $和$ \ Mathbb {t}^2 $）。然后，我们的分析在离散对称性下，在任何这样平滑表面上，全球平滑溶液的轨迹和无限时间丧失的常规性丧失。
We study the long-time behavior of scale-invariant solutions of the 2d Euler equation satisfying a discrete symmetry. We show that all scale-invariant solutions with bounded variation on $\mathbb{S}^1$ relax to states that are piece-wise constant with finitely many jumps. All continuous scale-invariant solutions become singular and homogenize in infinite time. On $\mathbb{R}^2$, this corresponds to generic infinite-time spiral and cusp formation. The main tool in our analysis is the discovery of a monotone quantity that measures the number of particles that are moving away from the origin. This monotonicity also applies locally to solutions of the 2d Euler equation that are $m$-fold symmetric ($m\geq 4$) and have radial limits at the point of symmetry. Our results are also applicable to the Euler equation on a large class of surfaces of revolution (like $\mathbb{S}^2$ and $\mathbb{T}^2$). Our analysis then gives generic spiraling of trajectories and infinite-time loss of regularity for globally smooth solutions on any such smooth surface, under a discrete symmetry.
