In this talk, 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. 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 flows, Burgers’ equations, and compressible Navier-Stokes equations, are presented. Finally, we successfully compute the control of conservation laws by incorporating both primal-dual algorithms and monotone schemes. This is based on a joint study with Prof. Osher and Siting Liu in the 2021 Thanksgiving holidays.
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