Nonsteady Navier-stokes Equations with Homogeneous Mixed Boundary Conditions

Let Ω be a bounded domain in IR with a Lipschitz boundary, ∂Ω ∈ | C and let Γ1, Γ2 be open disjoint subsets of ∂Ω such that ∂Ω = Γ1 ∪ Γ2, Γ1 6= ∅ and the 1-dimensional measure of ∂Ω − (Γ1 ∪ Γ2) is zero. The domain Ω represents a channel filled up with a fluid, Γ1 is a fixed wall and Γ2 is both the input and the output of the channel. Let T ∈ (0,∞], Q = Ω×(0, T ). The classical formulation of our problem is as follows: