Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films
For the thin-film model of a viscous flow which originates from lubrication approximation and has a full nonlinear curvature term, we prove existence of nonnegative weak solutions. Depending on initial data, we show algebraic or exponential dissipation of an energy functional which implies dissipati...
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| Format: | Article |
| Language: | English |
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Wiley
2016-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2016/4063740 |
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| _version_ | 1832567223149920256 |
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| author | Matthew Michal Marina Chugunova Roman Taranets |
| author_facet | Matthew Michal Marina Chugunova Roman Taranets |
| author_sort | Matthew Michal |
| collection | DOAJ |
| description | For the thin-film model of a viscous flow which originates from lubrication approximation and has a full nonlinear curvature term, we prove existence of nonnegative weak solutions. Depending on initial data, we show algebraic or exponential dissipation of an energy functional which implies dissipation of the solution arc length that is a well known property for a Hele-Shaw flow. For the classical thin-film model with linearized curvature term, under some restrictions on parameter and gradient values, we also prove analytically the arc length dissipation property for positive solutions. We compare the numerical solutions for both models, with nonlinear and with linearized curvature terms. In regimes when solutions develop finite time singularities, we explain the difference in qualitative behaviour of solutions. |
| format | Article |
| id | doaj-art-38219f50aad94ce0974eb527c4210605 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-38219f50aad94ce0974eb527c42106052025-02-03T01:02:02ZengWileyInternational Journal of Differential Equations1687-96431687-96512016-01-01201610.1155/2016/40637404063740Qualitative Behaviour of Solutions in Two Models of Thin Liquid FilmsMatthew Michal0Marina Chugunova1Roman Taranets2Institute of Mathematical Sciences, Claremont Graduate University, 710 N. College Avenue, Claremont, CA 91711, USAInstitute of Mathematical Sciences, Claremont Graduate University, 710 N. College Avenue, Claremont, CA 91711, USADepartment of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095, USAFor the thin-film model of a viscous flow which originates from lubrication approximation and has a full nonlinear curvature term, we prove existence of nonnegative weak solutions. Depending on initial data, we show algebraic or exponential dissipation of an energy functional which implies dissipation of the solution arc length that is a well known property for a Hele-Shaw flow. For the classical thin-film model with linearized curvature term, under some restrictions on parameter and gradient values, we also prove analytically the arc length dissipation property for positive solutions. We compare the numerical solutions for both models, with nonlinear and with linearized curvature terms. In regimes when solutions develop finite time singularities, we explain the difference in qualitative behaviour of solutions.http://dx.doi.org/10.1155/2016/4063740 |
| spellingShingle | Matthew Michal Marina Chugunova Roman Taranets Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films International Journal of Differential Equations |
| title | Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films |
| title_full | Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films |
| title_fullStr | Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films |
| title_full_unstemmed | Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films |
| title_short | Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films |
| title_sort | qualitative behaviour of solutions in two models of thin liquid films |
| url | http://dx.doi.org/10.1155/2016/4063740 |
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