2023/01/24 by Gladbach, Peter, Jansen, Jonas, Lienstromberg, Christina
#35B40 #35D30 #35K35 #35K65 #35Q35 #76A05 #76A20 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2301.10300
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 ∂t u + ∂x(m(u) |∂x3 u - G′′(u) ∂x u|α-1 (∂x3 u - G′′(u) ∂x u)) = 0 with potential G in the shear-thinning (α> 1), Newtonian (α= 1) and shear-thickening case (0 <α< 1). The latter satisfy an energy-dissipation inequality. Finally, we derive dissipation bounds in the case G≡ 0 which imply that solutions emerging from initial values with low energy lift up uniformly in finite time.