Title: A Unified Framework for Total Variation Regularized Optimization in Fluid Dynamics and Related Physical Systems

Abstract: An optimization framework is presented for minimizing the energy functional developed around a generalized equation governing physical systems such as fluid dynamics, particle transport, phase transition, and other related systems. The convexity of the energy functional is investigated to derive the necessary conditions for a smooth and global optimum solution. Furthermore, the Total Variation (TV) regularization term is introduced to gain insights into the solution space and convergence analysis of convection-dominated problems.
We demonstrate the practical application of our method by applying it to some selected examples such as the Boltzmann, Navier-Stokes, and Maxwell equations.