Doubly Nonlinear Thin-Film Equations in One Space Dimension

Smooth zero-contact-angle solutions to a thin-film equation around the steady state

“…We refer to [3,14] for up-todate reviews of the analytical theory (see also [8,21] for the partial wetting case). On the other hand, only quite partial results are available concerning regularity: in particular, a weak solution is known to satisfy (4) only for almost every time, and seems not to be strong enough to infer (2) (though its support is known to stay bounded, see [15] and the references therein).…”

New dissipated energies for the thin fluid film equation

“…Uniqueness is currently an open problem (except when the solution is positive [8]). See also the remarks below on the doubly nonlinear equation, for which a "p-Laplace" regularization is deployed in [2].…”

“…A rigorous derivation of the lubrication approximation (n = 3) is in [17]. An extension to non-Newtonian power-law fluids is treated in [2].…”

Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity

“…In the case when p is a constant, Model (1.1) have a sharp physical background and a rich theoretical significance. For example, this model may describe the surface tension driven evolution of the height u(x, t) of a thin liquid film on a solid surface in lubrication approximation [1,2]. Especially p ≡ 2, Problem (1.1) becomes the classical Cahn-Hilliard problem, and there have been many results about the existence, uniqueness, and some other properties of the solutions, the readers may refer to the bibliography given in [3][4][5].…”