We study a class of variational problems for regularized conservation laws with Lax's entropy-entropy flux pairs. We first introduce a modified optimal transport space based on conservation laws with diffusion. Using this space, we demonstrate that conservation laws with diffusion are "flux--gradient flows". We next construct variational problems for these flows, for which we derive dual PDE systems for regularized conservation laws. Several examples, including traffic flow and Burgers' equation, are presented. Incorporating both primal-dual algorithms and monotone schemes, we successfully compute the control of conservation laws.
翻译:我们用Lax的昆虫-昆虫通量对等体研究成常规养护法的变异问题。 我们首先根据扩散的养护法引入了经修改的最佳运输空间。 利用这个空间,我们证明扩散的养护法是“ 奢侈- 渐变流动 ” 。 我们接下来为这些流动构筑变异问题, 我们为此为规范养护法制定双重的PDE系统。 列举了几个例子, 包括交通流量和Burgers的等式。 将原始- 双轨算法和单调法都纳入其中, 我们成功地计算了保护法的控制权。