Mathematical and numerical analysis to the assembly of vortices in fluid

(1)我们考虑五点涡的顶点和中心的矩形。我们已经知道，对于初始形状和涡强度的某些参数，涡存在周期运动。我们研究另一个参数值的周期运动。结果发现，存在一族周期运动。通过数值模拟，这些周期运动具有简单的结构。(2)我们考虑五个点涡的顶点和中心的钻石。对于一定的涡旋强度值，五个涡旋处于相对平衡状态，即相对于具有均匀角速度的旋转坐标处于平衡状态。这个问题有两个参数。我们研究了相对平衡点的稳定性。通过数值模拟，我们发现平衡点在一个很窄的参数范围内是稳定的。为了证明稳定性，我们构造了Lyapunouv函数，并使用区间计算应用计算机辅助证明。结果表明，在相同的强度签名下，许多椭圆型平衡点是稳定的。到目前为止，我们没有成功地证明所有椭圆平衡的稳定性。(3)在（2）中所述的相同情况下，我们通过数值计算发现，某些不稳定的平衡点表现出弛豫振荡。我们不知道是否所有的不稳定平衡都表现出振荡，然而，我们的数值模拟表明，存在一个不表现出振荡的不稳定平衡。我们已经知道有正方形的五点涡表现出振荡，然而，我们发现的振荡属于另一种类型。这种振荡发生的数学原因是我们未来的工作之一。

(1) We consider five point vortices on the vertices and center of a rectangular. We already known that there exists a periodic motion of vortices for some parameters on the initial configuration and strength of vortices. We study the periodic motion for another value of parameters. As a result, we find that there exists a family of periodic motions. By numerical simulations, these periodic motions have a simple structure.(2) We consider five point vortices on the vertices and center of a diamond. For some values of strength of vortices, the five vortices is in a relative equilibrium, that is, it is in a equilibrium with respect to a rotation coordinate with uniform angular velocity. There are two parameters in this problem. We study the stability of the relative equilibrium. By numerical simulations, we find that the equilibria are stable in a narrow parameter range. To show the stability, we construct the Lyapunouv function and apply the computer-assisted proof using interval computations. As a result, under the same signature of strengths, we find that many elliptic equilibria are stable. Until now we do not succeed in proving stability for all elliptic equilibria.(3) Under the same situation stated in (2), we numerically find that some unstable equilibria exhibit the relaxation oscillation. We do not know whether or not all unstable equilibria exhibit the oscillation, however, our numerical simulations suggest that an unstable equilibrium which does not exhibit the oscillation exists. We already know that there are square-shaped five point vortices which exhibit the oscillation, however, the oscillation we find belongs to a different kind category. The mathematical reason why such a oscillation occurs is one of our future works.