On the Stability of Laminar Flows Between Plates

Consider a two-dimensional laminar flow between two plates, so that $$(x_1,x_2)\in {{\mathbb {R}}}\times [-1,1]$$ , given by $${{\mathbf {v}}}(x_1,x_2)=(U(x_2),0)$$ where $$U\in C^4([-1,1])$$ satisfies $$U^\prime \ne 0$$ in $$[-1,1]$$ . We prove the is linearly stable large Reynolds number limit, different cases: assume either no-slip or fixed traction force (Navier-slip) conditions on and an arbitrary (but much smaller than number) period $$x_1$$ direction.