论文标题
维度1+1中波图方程的统计力学
Statistical mechanics of the wave maps equation in dimension 1+1
论文作者
论文摘要
我们在s^d中使用值研究的波图，在未来的光锥{| x |中定义<= t}，在边界处有规定的数据{| x | = t}。根据Keel和Tao的工作，我们证明了问题在本地绝对连续的边界数据中得到了很好的解决。我们设计了问题的离散版本，并证明，对于每个绝对连续的边界数据，随着网格大小趋向于0，离散问题的解决方案的顺序会收敛到相应的连续波图。我们证明，离散问题的解决方案序列具有局部均匀收敛拓扑的累积点。我们认为所产生的随机场可以解释为与Gibbs分布给出的初始数据相对应的波形演化。
We study wave maps with values in S^d, defined on the future light cone {|x| <= t}, with prescribed data at the boundary {|x| = t}. Based on the work of Keel and Tao, we prove that the problem is well-posed for locally absolutely continuous boundary data. We design a discrete version of the problem and prove that for every absolutely continuous boundary data, the sequence of solutions of the discretised problem converges to the corresponding continuous wave map as the mesh size tends to 0. Next, we consider the boundary data given by the S^d-valued Brownian motion. We prove that the sequence of solutions of the discretised problems has an accumulation point for the topology of locally uniform convergence. We argue that the resulting random field can be interpreted as the wave-map evolution corresponding to the initial data given by the Gibbs distribution.
