Time-dependent coating flows in a strip, Part I: The linearized problem

This work is concerned with time-dependent coating flow in a strip $0 < y < 1$. The Navier-Stokes equations are satisfied in the fluid region, the bottom substrate $y = 0$ is moving with fixed velocity $(U,0)$, and fluid is entering the strip through the upper boundary $y = 1$. The free boundary has the form $y = f(x,t)$ for $-\infty < x < R(t)$, where $R(t)$ is the moving contact point. Our objective is to prove that if the initial data are close to those of a stationary solution (the existence of such a solution was established by the authors in an earlier paper) then the time-dependent problem has a unique solution with smooth free boundary, at least for a small time interval. In this Part I we study the linearized problem, about the stationary solution, and obtain sharp estimates for the solution and its derivatives. These estimates will be used in Part II to establish existence and uniqueness for the full nonlinear problem.