The inverse initial data problem for anisotropic Navier-Stokes equations via Legendre time reduction method

We consider the inverse initial data problem for the compressible anisotropic Navier-Stokes equations, where the goal is to reconstruct the initial velocity field from lateral boundary observations. This problem arises in applications where direct measurements of internal fluid states are unavailable. We introduce a novel computational framework based on Legendre time reduction, which projects the velocity field onto an exponentially weighted Legendre basis in time. This transformation reduces the original time-dependent inverse problem to a coupled, time-independent elliptic system. The resulting reduced model is solved iteratively using a Picard iteration and a stabilized least-squares formulation under noisy boundary data. Numerical experiments in two dimensions confirm that the method accurately and robustly reconstructs initial velocity fields, even in the presence of significant measurement noise and complex anisotropic structures. This approach offers a flexible and computationally tractable alternative for inverse modeling in fluid dynamics with anisotropic media.