Print Email Facebook Twitter Stability of receding traveling waves for a fourth order degenerate parabolic free boundary problem Title Stability of receding traveling waves for a fourth order degenerate parabolic free boundary problem Author Gnann, M.V. (TU Delft Analysis) Ibrahim, Slim (University of Victoria) Masmoudi, Nader (New York University) Date 2019 Abstract Consider the thin-film equation h _{t} +(hh _{yyy} ) _{y} =0 with a zero contact angle at the free boundary, that is, at the triple junction where liquid, gas, and solid meet. Previous results on stability and well-posedness of this equation have focused on perturbations of equilibrium-stationary or self-similar profiles, the latter eventually wetting the whole surface. These solutions have their counterparts for the second-order porous-medium equation h _{t} −(h ^{m} ) _{yy} =0, where m>1 is a free parameter. Both porous-medium and thin-film equation degenerate as h↘0, but the porous-medium equation additionally fulfills a comparison principle while the thin-film equation does not. In this note, we consider traveling waves h=[Formula presented]x ^{3} +νx ^{2} for x≥0, where x=y−Vt and V,ν≥0 are free parameters. These traveling waves are receding and therefore describe de-wetting, a phenomenon genuinely linked to the fourth-order nature of the thin-film equation and not encountered in the porous-medium case as it violates the comparison principle. The linear stability analysis leads to a linear fourth-order degenerate-parabolic operator for which we prove maximal-regularity estimates to arbitrary orders of the expansion in x in a right-neighborhood of the contact line x=0. This leads to a well-posedness and stability result for the corresponding nonlinear equation. As the linearized evolution has different scaling as x↘0 and x→∞ the analysis is more intricate than in related previous works. We anticipate that our approach is a natural step towards investigating other situations in which the comparison principle is violated, such as droplet rupture. Subject Degenerate-parabolic equationsFourth-order equationFree boundary problemsStabilityThin film equationTraveling waves To reference this document use: http://resolver.tudelft.nl/uuid:115e8e71-32d1-4ca3-b816-f95c01a8b351 DOI https://doi.org/10.1016/j.aim.2019.01.028 Embargo date 2021-03-22 ISSN 0001-8708 Source Advances in Mathematics, 347, 1173-1243 Bibliographical note Accepted author manuscript Part of collection Institutional Repository Document type journal article Rights © 2019 M.V. Gnann, Slim Ibrahim, Nader Masmoudi Files PDF Gnann_Ibrahim_Masmoudi_2018.pdf 1018.34 KB Close viewer /islandora/object/uuid:115e8e71-32d1-4ca3-b816-f95c01a8b351/datastream/OBJ/view