TY - JOUR
T1 - Stability of receding traveling waves for a fourth order degenerate parabolic free boundary problem
AU - Gnann, Manuel V.
AU - Ibrahim, Slim
AU - Masmoudi, Nader
N1 - Funding Information:
MVG appreciates discussions with Mircea Petrache on a related problem. The authors acknowledge funding from and the kind hospitality of the Fields Institute for Research in Mathematical Sciences in Toronto and the New York University in Abu Dhabi. MVG and SI are grateful to the Courant Institute of the New York University in New York City for hosting them. MVG and NM wish to thank the University of Victoria, BC, for its kind hospitality. MVG also received funding from the University of Michigan at Ann Arbor, National Science Foundation (NSF) grant DMS-1054115, the Max Planck Institute for Mathematics in the Sciences in Leipzig, and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project # 334362478. SI was partially supported by National Sciences and Engineering Research Council (NSERC) Discovery grant # 371637-2014, and NM was in part supported by National Science Foundation (NSF) grant DMS-1211806 and DMS-1716466. The authors thank the anonymous reviewer for several suggestions that have lead to an improved presentation of this revised version.☆ MVG appreciates discussions with Mircea Petrache on a related problem. The authors acknowledge funding from and the kind hospitality of the Fields Institute for Research in Mathematical Sciences in Toronto and the New York University in Abu Dhabi. MVG and SI are grateful to the Courant Institute of the New York University in New York City for hosting them. MVG and NM wish to thank the University of Victoria, BC, for its kind hospitality. MVG also received funding from the University of Michigan at Ann Arbor, National Science Foundation (NSF) grant DMS-1054115, the Max Planck Institute for Mathematics in the Sciences in Leipzig, and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project # 334362478. SI was partially supported by National Sciences and Engineering Research Council (NSERC) Discovery grant # 371637-2014, and NM was in part supported by National Science Foundation (NSF) grant DMS-1211806 and DMS-1716466. The authors thank the anonymous reviewer for several suggestions that have lead to an improved presentation of this revised version.
Publisher Copyright:
© 2019 Elsevier Inc.
PY - 2019/4/30
Y1 - 2019/4/30
N2 - 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.
AB - 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.
KW - Degenerate-parabolic equations
KW - Fourth-order equation
KW - Free boundary problems
KW - Stability
KW - Thin film equation
KW - Traveling waves
UR - http://www.scopus.com/inward/record.url?scp=85063225242&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85063225242&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2019.01.028
DO - 10.1016/j.aim.2019.01.028
M3 - Article
AN - SCOPUS:85063225242
SN - 0001-8708
VL - 347
SP - 1173
EP - 1243
JO - Advances in Mathematics
JF - Advances in Mathematics
ER -