2019/03/16 by Hangjie Ji, Ji, Hangjie, Thomas P. Witelski +1 · 1 citation
Computer Science · Engineering · Materials Science · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Thin Films #Nonlinear Dynamics and Pattern Formation #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.1903.06987
openalex publication_date 2019/03/16 · openalex created_date 2019/11/01 · openalex updated_date 2026/07/28
We study the steady states and dynamics of a thin film-type equation with non-conserved mass in one dimension. The evolution equation is a nonlinear fourth-order degenerate parabolic PDE motivated by a model of volatile viscous fluid films allowing for condensation or evaporation. We show that by changing the sign of the non-conserved flux and breaking from a gradient flow structure, the problem can exhibit novel behaviors including having two distinct classes of coexisting steady state solutions. Detailed analysis of the bifurcation structure for these steady states and their stability reveal several possibilities for the dynamics. For some parameter regimes, solutions can lead to finite-time rupture singularities. Interestingly, we also show that a finite amplitude limit cycle can occur as a singular perturbation in the nearly-conserved limit.