Abstract: In this paper we discuss the mixed boundary problem (1)-(3) of the Navier-Stokes equations for the flow of an incompressible viscous fluid in a bounded domain. We prove that when g∈L^∞(Q),φ∈C¹(Γ₁×[0,t₁]),and ψ∈L¹(Γ₂×[0,t₁]),there exists a weak solution of (1)-(3),and when u∈L²(0,t₁;H³(Ω)),T∈L²(0,t₁;H³(Ω)),the weak solution is unique,if it exists.

Key words: Navier-Stokes equations, weak solution, mixed boundary poblem