Abstract: For the generalized stochastic Navier-Stokes equations with Navier boundary condition,we prove a priori estimates for the Navier-Stokes equations in L² and L⁴.And then,we prove the monotonicity of nonlinear terms.At last by using Minty-Browder we get the global existence and uniqueness for the weak solution of the stochastic equation.

Key words: Navier slip boundary, damping, stochastic Navier-Stokes equation, well-posedness