Non-steady Navier-Stokes equations with homogeneous mixed boundary conditions and arbitrarily large initial condition

Let Ω ⊂ R2 be a bounded domain, ∂Ω ∈ C0,1 and ∂Ω = Γ1 ∪ Γ2 such that Γ1 and Γ2 are closed, sufficiently smooth, 1-dimensional measure of Γ1 ∩ Γ2 is zero and 1-dimensional measure of Γ1 is positive. Further let (0, T) be a time interval. We prescribe the non-slip boundary conditions on Γ1 × (0, T) and the boundary condition −Pn + ∂u ∂n = 0 on Γ2 × (0, T). Here u = (u1, u2) is velocity, P represents pressure and n = (n1, n2) is an outer normal vector. Our aim is to prove the existence and uniqueness of this problem on some time interval (0, T ∗) for sufficiently small T ∗, 0 < T ∗ ≤ T.