Abstract
We use Feireisl-Lions theory to deduce the existence of weak solutions to a system describing the dynamics of a linear oscillator containing a Newtonian compressible fluid. The appropriate Navier-Stokes equation is considered on a domain whose movement has one degree of freedom. The equation is paired with the Newton law and we assume a no-slip boundary condition.
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Notes
Note that \(({{\,\mathrm{div}\,}}(\varrho \mathbf{u}\otimes \mathbf{v}))_j = \partial _i(\varrho \mathbf{v}_i \mathbf{u}_j)\) where the summation convention is used.
Hereinafter we use an abbreviation \({\max \{8,\gamma \}} = \beta\).
Hereinafter we use the following notation: \(\Vert \cdot \Vert _{L^q}\) is a norm in \(L^q(\Omega )\), \(\Vert \cdot \Vert _{W^{k,q}}\) is a norm in \(W^{k,q}(\Omega )\), \(\Vert \cdot \Vert _{L^qL^r}\) is norm in \(L^q((0,T),L^r(\Omega ))\) and \(\Vert \cdot \Vert _{L^qW^{k,r}}\) is a norm in \(L^q((0,T),W^{k,r}(\Omega ))\).
Here \(\rightrightarrows\) denotes the uniform convergence.
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This research was funded by Czech Science Foundation project number Grant GA19-04243S in the framework of RVO: 67985840.
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Mácha, V. Compressible fluid inside a linear oscillator. J Elliptic Parabol Equ 7, 393–416 (2021). https://doi.org/10.1007/s41808-021-00120-1
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DOI: https://doi.org/10.1007/s41808-021-00120-1