Weak solutions and singular limits for a compressible fluid-structure interaction problem with slip boundary conditions
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arXiv
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| Natura: | Preprint |
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2024
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| author | Liu, Yadong Mitra, Sourav Nečasová, Šárka |
| author_facet | Liu, Yadong Mitra, Sourav Nečasová, Šárka |
| contents | We study a system describing the compressible barotropic fluids interacting with (visco) elastic solid shell/plate. In particular, the elastic structure is part of the moving boundary of the fluid, and the Navier-slip type boundary condition is taken into account. Depending on the reference geometry (flat or not), we show the existence of weak solutions to the coupled system provided the adiabatic exponent satisfies $γ> \frac{12}{7}$ without damping and $γ> \frac{3}{2}$ with structure damping, utilizing the domain extension and regularization approximation. Moreover, via a modified relative entropy method in time-dependent domains, we give a rigorous justification of the incompressible inviscid limit of the compressible fluid-structure interaction problem with a flat reference geometry, in the regime of low Mach number, high Reynolds number, and well-prepared initial data. As a byproduct, with a fixed Reynolds number, we derive the incompressible limit without extra assumption. To the best of our knowledge, this is the first result concerning the singular limit problem for compressible fluids interacting with elastic structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09908 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak solutions and singular limits for a compressible fluid-structure interaction problem with slip boundary conditions Liu, Yadong Mitra, Sourav Nečasová, Šárka Analysis of PDEs 76N10, 74F10, 35Q35, 35R37, 35B25 We study a system describing the compressible barotropic fluids interacting with (visco) elastic solid shell/plate. In particular, the elastic structure is part of the moving boundary of the fluid, and the Navier-slip type boundary condition is taken into account. Depending on the reference geometry (flat or not), we show the existence of weak solutions to the coupled system provided the adiabatic exponent satisfies $γ> \frac{12}{7}$ without damping and $γ> \frac{3}{2}$ with structure damping, utilizing the domain extension and regularization approximation. Moreover, via a modified relative entropy method in time-dependent domains, we give a rigorous justification of the incompressible inviscid limit of the compressible fluid-structure interaction problem with a flat reference geometry, in the regime of low Mach number, high Reynolds number, and well-prepared initial data. As a byproduct, with a fixed Reynolds number, we derive the incompressible limit without extra assumption. To the best of our knowledge, this is the first result concerning the singular limit problem for compressible fluids interacting with elastic structures. |
| title | Weak solutions and singular limits for a compressible fluid-structure interaction problem with slip boundary conditions |
| topic | Analysis of PDEs 76N10, 74F10, 35Q35, 35R37, 35B25 |
| url | https://arxiv.org/abs/2405.09908 |