论文标题
系统地搜索3D Euler流中的奇异点
Systematic search for singularities in 3D Euler flows
论文作者
论文摘要
我们考虑一个问题，是否从平滑的初始条件开始3D Inviscid Euler在周期域上流动$ \ Mathbb {t}^3 $可能会在有限的时间内发展出奇异性。我们的出发点是Kato（1972）的众所周知的结果，该结果主张了Sobolev Space $ H^M（\ Mathbb {T}^3）$ o $ m> 5/2 $中的Euler系统的当地存在。因此，随着奇异时间的接近，奇异性的潜在形成必须伴随着速度场的$ h^m $规范的无限生长。我们对“极端” Euler流进行系统的搜索，该流程可能通过制定和解决PDE约束优化问题来实现这种情况，在某个固定时间$ h^3 $ norma n Norm $ h^3 $规范在某个固定时间$ t> 0 $相对于初始数据最大化，但对适当的归一化约束约束。使用最先进的Riemannian共轭梯度方法来解决此问题，其中从伴随系统的溶液中获得梯度。 Computations performed with increasing numerical resolutions demonstrate that, as asserted by the theorem of Kato (1972), when the optimization time window $[0, T]$ is sufficiently short, the $H^3$ norm remains bounded in the extreme flows found by solving the optimization problem, which indicates that the Euler system is well-posed on this "short" time interval.另一方面，当窗口$ [0，t] $长，可能比Kato定理所主张的本地存在时间更长，然后在分辨率细化时极端流动的$ H^3 $ norm差异，这表明在这个“长”时间间隔上可能的奇异性表述。长时间窗口中获得的极端流具有两个碰撞涡流环的形式，并以某些对称性为特征。特别是，可能发生奇异性的流量几乎是轴对称的。
We consider the question whether starting from a smooth initial condition 3D inviscid Euler flows on a periodic domain $\mathbb{T}^3$ may develop singularities in a finite time. Our point of departure is the well-known result by Kato (1972), which asserts the local existence of classical solutions to the Euler system in the Sobolev space $H^m(\mathbb{T}^3)$ for $m > 5/2$. Thus, potential formation of a singularity must be accompanied by an unbounded growth of the $H^m$ norm of the velocity field as the singularity time is approached. We perform a systematic search for "extreme" Euler flows that may realize such a scenario by formulating and solving a PDE-constrained optimization problem where the $H^3$ norm of the solution at a certain fixed time $T > 0$ is maximized with respect to the initial data subject to suitable normalization constraints. This problem is solved using a state-of-the-art Riemannian conjugate gradient method where the gradient is obtained from solutions of an adjoint system. Computations performed with increasing numerical resolutions demonstrate that, as asserted by the theorem of Kato (1972), when the optimization time window $[0, T]$ is sufficiently short, the $H^3$ norm remains bounded in the extreme flows found by solving the optimization problem, which indicates that the Euler system is well-posed on this "short" time interval. On the other hand, when the window $[0, T]$ is long, possibly longer than the time of the local existence asserted by Kato's theorem, then the $H^3$ norm of the extreme flows diverges upon resolution refinement, which indicates a possible singularity formulation on this "long" time interval. The extreme flow obtained on the long time window has the form of two colliding vortex rings and is characterized by certain symmetries. In particular, the region of the flow in which a singularity might occur is nearly axisymmetric.
