{"6542ad7fbcea229e7f345790a76dbc7felkepeter":{"DOI":"10.48550/ARXIV.2301.10300","ISBN":"","ISSN":"","URL":"https://arxiv.org/abs/2301.10300","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\r\n∂tu+∂x(m(u)|∂3xu−G′′(u)∂xu|α−1(∂3xu−G′′(u)∂xu))=0\r\nwith 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.","annote":"","author":[{"family":"Gladbach","given":"Peter"},{"family":"Jansen","given":"Jonas"},{"family":"Lienstromberg","given":"Christina"}],"citation-label":"https://doi.org/10.48550/arxiv.2301.10300","collection-editor":[],"collection-title":"","container-author":[],"container-title":"","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2023"]],"literal":"2023"},"event-place":"","id":"6542ad7fbcea229e7f345790a76dbc7felkepeter","interhash":"939b1f264055663e904645113e2b2e05","intrahash":"6542ad7fbcea229e7f345790a76dbc7f","issue":"","issued":{"date-parts":[["2023"]],"literal":"2023"},"keyword":"Lienstromberg Analysis IADM PDEs gradient-flow FOS","misc":{"copyright":"arXiv.org perpetual, non-exclusive license","doi":"10.48550/ARXIV.2301.10300"},"note":"","number":"","page":"","page-first":"","publisher":"arXiv","publisher-place":"","status":"","title":"Non-Newtonian thin-film equations: global existence of solutions, gradient-flow structure and guaranteed lift-off","type":"article-journal","username":"elkepeter","version":"","volume":""}}