标题： Ginzburg-Landau涡旋在曲面上矢量场的动力学 显示英文标题

标题： Dynamics of Ginzburg-Landau vortices for vector fields on surfaces

摘要： 在本文中，我们考虑以下Ginzburg-Landau型能量的梯度流\[ F_\varepsilon(u) := \frac{1}{2}\int_{M}\vert D u\vert_g^2 +\frac{1}{2\varepsilon^2}\left(\vert u\vert_g^2-1\right)^2\mathrm{vol}_g. \] 该能量定义在维数为$2$的闭合且定向的黎曼流形$M$上的切向量场（此处$D$表示协变导数），并且依赖于一个小型参数$\varepsilon>0$。 如果能量满足适当的界限，当$\varepsilon\to 0$时，第二项会使向量场具有单位长度。 然而，由于在 Sobolev 正则性和单位范数约束之间向量场在$M$上的不相容性，$F_\varepsilon$的临界点倾向于生成有限数量的奇点（称为涡旋），这些奇点具有非零指标（当欧拉特征非零时）。这类问题最近由 R. Ignat 和 R. Jerrard 在一篇论文中进行了广泛分析。与欧几里得情况类似，涡旋的位置由所谓的重整化能量所决定。在本文中，我们关注涡旋的动力学。我们严格证明了涡旋按照重整化能量的梯度流移动，这是当$\varepsilon\to 0$时 Ginzburg-Landau 能量梯度流的极限行为。 显示英文摘要

摘要： In this paper we consider the gradient flow of the following Ginzburg-Landau type energy \[ F_\varepsilon(u) := \frac{1}{2}\int_{M}\vert D u\vert_g^2 +\frac{1}{2\varepsilon^2}\left(\vert u\vert_g^2-1\right)^2\mathrm{vol}_g. \] This energy is defined on tangent vector fields on a $2$-dimensional closed and oriented Riemannian manifold $M$ (here $D$ stands for the covariant derivative) and depends on a small parameter $\varepsilon>0$. If the energy satisfies proper bounds, when $\varepsilon\to 0$ the second term forces the vector fields to have unit length. However, due to the incompatibility for vector fields on $M$ between the Sobolev regularity and the unit norm constraint, critical points of $F_\varepsilon$ tend to generate a finite number of singular points (called vortices) having non-zero index (when the Euler characteristic is non-zero). These types of problems have been extensively analyzed in a recent paper by R. Ignat and R. Jerrard. As in Euclidean case, the position of the vortices is ruled by the so-called renormalized energy. In this paper we are interested in the dynamics of vortices. We rigorously prove that the vortices move according to the gradient flow of the renormalized energy, which is the limit behavior when $\varepsilon\to 0$ of the gradient flow of the Ginzburg-Landau energy.