Existence and uniqueness for a coupled parabolic-hyperbolic model of MEMS
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914743072063488 |
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| author | Gimperlein, Heiko He, Runan Lacey, Andrew A. |
| author_facet | Gimperlein, Heiko He, Runan Lacey, Andrew A. |
| contents | Local well-posedness for a nonlinear parabolic-hyperbolic coupled system modelling Micro-Electro-Mechanical System (MEMS) is studied. The particular device considered is a simple capacitor with two closely separated plates, one of which has motion modelled by a semi-linear hyperbolic equation. The gap between the plates contains a gas and the gas pressure is taken to obey a quasi-linear parabolic Reynolds' equation. Local-in-time existence of strict solutions of the system is shown, using well-known local-in-time existence results for the hyperbolic equation, then Hölder continuous dependence of its solution on that of the parabolic equation, and finally getting local-in-time existence for a combined abstract parabolic problem. Semigroup approaches are vital for the local-in-time existence results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04128 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence and uniqueness for a coupled parabolic-hyperbolic model of MEMS Gimperlein, Heiko He, Runan Lacey, Andrew A. Analysis of PDEs Local well-posedness for a nonlinear parabolic-hyperbolic coupled system modelling Micro-Electro-Mechanical System (MEMS) is studied. The particular device considered is a simple capacitor with two closely separated plates, one of which has motion modelled by a semi-linear hyperbolic equation. The gap between the plates contains a gas and the gas pressure is taken to obey a quasi-linear parabolic Reynolds' equation. Local-in-time existence of strict solutions of the system is shown, using well-known local-in-time existence results for the hyperbolic equation, then Hölder continuous dependence of its solution on that of the parabolic equation, and finally getting local-in-time existence for a combined abstract parabolic problem. Semigroup approaches are vital for the local-in-time existence results. |
| title | Existence and uniqueness for a coupled parabolic-hyperbolic model of MEMS |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.04128 |