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We study the gradient-flow structure of a non-Newtonian thin film equationwith power-law rheology. The equation is quasilinear, of fourth order anddoubly-degenerate parabolic. By adding a singular potential to the naturalDirichlet energy, we introduce a modified version of the thin-film equation.Then, we set up a minimising-movement scheme that converges to global positiveweak solutions to the modified problem. These solutions satisfy anenergy-dissipation equality and follow a gradient flow. In the limit of avanishing singularity of the potential, we obtain global non-negative weaksolutions 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_xu\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 derivedissipation bounds in the case $G\equiv 0$ which imply that solutions emergingfrom initial values with low energy lift up uniformly in finite time.

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