The thin film equation with backwards second order diffusion

Abstract

We focus on the thin film equation with lower order "backwards" diffusion which can describe, for example, the evolution of thin viscous films in the presence of gravity and thermo-capillary effects, or the thin film equation with a "porous media cutoff" of van der Waals forces. We treat in detail the equationwhere ν = ±1, n > 0, M > m, and A 0. Global existence of weak nonnegative solutions is proven when m − n > −2 and A > 0 or ν = −1, and when −2 < m − n < 2, A = 0, ν = 1. From the weak solutions, we get strong entropy solutions under the additional constraint that m−n > −3/2 if ν = 1. A local energy estimate is obtained when 2 n < 3 under some additional restrictions. Finite speed of propagation is proven when m > n/2, for the case of "strong slippage", 0 < n < 2, when ν = 1 based on local entropy estimates, and for the case of "weak slippage", 2 n < 3, when ν = ±1 based on local entropy and energy estimates.

Initial propagation of support in thin-film flow

“…In [13] by FI-method for thin-film equation with backward diffusion (Equation (4) with l = 0), the FSP-property was proved for 0 < n < 3, m > n 2 . In [14] for thin-film equation with nonlinear convection…”

Finite-time thin film rupture driven by modified evaporative loss

“…corresponding to the leading order PDE This is a thin-film type equation with stabilizing di↵usion [40], where both nonlinear terms work to smooth the solution and prevent rupture. The positivity of solutions for all t > 0 can be demonstrated by means of the Theorem from [11], applied with U given by U (h) = 1 3 h 3 .…”

Existence and stability of solutions for a strongly coupled system modelling thin fluid films

“…We note that when one of the functions f or h is constantly equal to zero, the other unknown is a solution of the thin film equation with nonlinear diffusion whose investigation has and still receives a lot of attention [3,10,[15][16][17]20]. In the absence of surface tension effects, that is when A = B = 0, problem (1.1) has been analyzed in [6] for strong solutions, and in [4] in the degenerate case when the two interfaces may enter in contact or touch the bottom.…”