Well-posedness and stability for a mixed order system arising in thin film equations with surfactant. (English) Zbl 1523.35041
Summary: The objective of the present work is to provide a well-posedness result for a capillary driven thin film equation with insoluble surfactant. The resulting parabolic system of evolution equations is not only strongly coupled and degenerated, but also of mixed orders. To the best of our knowledge the only well-posedness result for a capillary driven thin film with surfactant is provided in [Nonlinear Anal., Real World Appl. 27, 124–145 (2016; Zbl 1331.76022)] by the same author, where a severe smallness condition on the surfactant concentration is assumed to prove the result. Thus, in spite of an intensive analytical study of thin film equations with surfactant during the last decade, a proper well-posedness result is still missing in the literature. It is the aim of the present paper to fill this gap. Furthermore, we apply a recently established result on asymptotic stability in interpolation spaces [B.-V. Matioc and C. Walker, “On the principle of linearized stability in interpolation spaces for quasilinear evolution equations”, Preprint, arXiv:1804.10523] to prove that the flat equilibrium of our system is asymptotically stable.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K52 | Initial-boundary value problems for higher-order parabolic systems |
| 35K59 | Quasilinear parabolic equations |
| 35K65 | Degenerate parabolic equations |
| 76A20 | Thin fluid films |
Keywords:
asymptotic stability; degenerate parabolic system; local well-posedness; mixed orders; thin film equationsCitations:
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