This paper is conced with the global well-posedness and regularity of weak solutions for the 2D non-autonomous incompressible Navier-Stokes equation with a inhomogeneous boundary condition in Lipschitz-like domain.Using the estimate for goving steady state equation and Hardy’s inequality,the existence and regularity of global unique weak solution can be proved.Moreover,these results also hold for 2D Navier-Stokes equation with Rayleigh’s friction and Navier-Stokes-Voigt flow,but invalid for three dimension.