Notes on Blowup and Long Wave Unstable Thin Film Equations

(1) ut = −(uuxxx)x − (uux)x. This is the one dimensional version of ut = −∇ · (f(u)∇∆u)−∇ · (g(u)∇u), with f(u) = un and g(u) = un. Such equations have been used to model the dynamics of a thin film of viscous liquid spreading on a flat solid surface. The air/liquid interface is at height z = u(x, y, t) ≥ 0 and the liquid/solid interface is at z = 0. The one dimensional equation (1) applies if the liquid film is uniform in the y direction. We refer to [16], [17] for reviews of the physical and modeling literature. Bertozzi and Pugh [4] introduced three regimes for the equation: subcritical (m < n + 2), critical (m = n + 2), and supercritical (m > n + 2). In these notes, we give an introduction to these dynamical regimes. Before addressing why the balance between m and n+2 is crucial, we first consider a more classical PDE which has analogous regimes. In addition, we refer the reader to Levine’s survey article on the role of critical exponents in blowup theorems [13].