In the present paper, we establish the well-posedness, stability, and (weak) convergence of a fully-discrete approximation of the unsteady $p(\cdot,\cdot)$-Navier-Stokes equations employing an implicit Euler step in time and a discretely inf-sup-stable finite element approximation in space. Moreover, numerical experiments are carried out that supplement the theoretical findings.

翻译：本文建立了采用时间方向上隐式Euler步进与空间方向上离散inf-sup稳定有限元逼近的非定常$p(\cdot,\cdot)$-Navier-Stokes方程全离散格式的适定性、稳定性及（弱）收敛性。此外，还通过数值实验补充验证了理论结果。