A Carleman contraction method for inverse initial data recovery in the Navier-Stokes equations with unknown body force

We solve an inverse initial data problem for the incompressible Navier-Stokes system. The objective is to recover the initial velocity and pressure from lateral boundary observations, without assuming that the time-independent body force is known. To eliminate this unknown force, we differentiate the momentum equation with respect to time and then apply a Legendre polynomial-exponential time-dimensional reduction. This procedure yields a coupled system of elliptic equations for the expansion coefficients. We then construct a contractive map for this reduced system on a suitable admissible set equipped with a Carleman-weighted norm. Its fixed point yields an approximate solution of the time-dimensional reduction model, and the contraction property gives rise to a globally convergent Picard iteration. Finally, we present a numerical algorithm based on this framework and numerical experiments showing accurate reconstructions of the initial velocity and pressure from synthetic boundary data.