Non-Newtonian thin-film equations: global existence of solutions, gradient-flow structure and guaranteed lift-off

Abstract

We study the gradient-flow structure of a non-Newtonian thin film equation with power-law rheology. The equation is quasilinear, of fourth order and doubly-degenerate parabolic. By adding a singular potential to the natural Dirichlet energy, we introduce a modified version of the thin-film equation. Then, we set up a minimising-movement scheme that converges to global positive weak solutions to the modified problem. These solutions satisfy an energy-dissipation equality and follow a gradient flow. In the limit of a vanishing singularity of the potential, we obtain global non-negative weak solutions to the power-law thin-film equation \begin{equation*} \partial_t u + \partial_x\bigl(m(u) |\partial_x^3 u - G^{\prime\prime}(u) \partial_x u|^{\alpha-1} \bigl(\partial_x^3 u - G^{\prime\prime}(u) \partial_x u\bigr)\bigr) = 0 \end{equation*} with potential $G$ in the shear-thinning ($\alpha > 1$), Newtonian ($\alpha = 1$) and shear-thickening case ($0 <\alpha < 1$). The latter satisfy an energy-dissipation inequality. Finally, we derive dissipation bounds in the case $G\equiv 0$ which imply that solutions emerging from initial values with low energy lift up uniformly in finite time.