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

We consider an inverse initial-data problem for the compressible anisotropic Navier--Stokes equations, in which the goal is to reconstruct the initial velocity field from noisy lateral boundary observations. In the formulation studied here, the density, pressure, anisotropic viscosity tensor, and body force are assumed known, while the initial velocity is the quantity to be recovered. We introduce a new computational framework based on Legendre time-dimensional reduction, in which the velocity field is projected onto an exponentially weighted Legendre basis in time. This transformation reduces the original time-dependent inverse problem to a coupled system of time-independent elliptic equations for the Fourier coefficients of the velocity field. The resulting reduced model is solved using a combination of quasi-reversibility and a damped Picard iteration. Numerical experiments in two dimensions show that the proposed method accurately and robustly reconstructs initial velocity fields, even in the presence of significant measurement noise, geometrically complex structures, and anisotropic effects. The method provides a flexible and computationally tractable approach for inverse fluid problems in anisotropic media.