Abstract
The principle purpose of this work is to investigate a “viscous” version of a “simple” but still realistic bi-fluid model described in Bresch et al. (in: Giga, Novotný (eds) Handbook of mathematical analysis in mechanics of viscous fluids, 2018) whose “non-viscous” version is derived from physical considerations in Ishii and Hibiki (Thermo-fluid dynamics of two-phase flow, Springer, Berlin, 2006) as a particular sample of a multifluid model with algebraic closure. The goal is to show the existence of weak solutions for large initial data on an arbitrarily large time interval. We achieve this goal by transforming the model to a transformed two-densities system which resembles the compressible Navier–Stokes equations, with, however, two continuity equations and a momentum equation endowed with the pressure of a complicated structure dependent on two variable densities. The new “transformed two-densities system” is then solved by an adaptation of the Lions–Feireisl approach for solving compressible Navier–Stokes equation, completed with several observations related to the DiPerna–Lions transport theory inspired by Maltese et al. (J Differ Equ 261:4448–4485, 2016) and Vasseur et al. (J Math Pures Appl 125:247–282, 2019). We also explain how these techniques can be generalized to a model of mixtures with more than two species. This is the first result on the existence of weak solutions for any realistic multifluid system.
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Notes
The domination property could be avoided provided P is growing in both variables sufficiently quickly at infinity (with power \(\ge 9/5\) in both variables), see the recent paper [23].
Note that \(d_{\pm }\) or \(e_{\pm }\) may blow up when \(r\rightarrow 0^+\).
Denoting by \(\mathbf{n}\) the outer normal to the boundary \(\partial {\mathscr {O}}_{{\underline{a}}}\), one can take for example \({{\tilde{P}}}(z)=P(\varrho ,{\underline{a}}\varrho )\) if z belongs to the half-line \(\{(\varrho ,{\underline{a}}\varrho ) +t\mathbf{n}(\varrho ,{\underline{a}}\varrho )|t>0\}\) with \(\varrho >0\), \({{\tilde{P}}}(z)=P(\varrho ,{{\overline{a}}}\varrho )\) if z belongs to the half-line \(\{(\varrho ,{{\overline{a}}}\varrho ) +t\mathbf{n}(\varrho ,{{\overline{a}}}\varrho )|t>0\}\) with \(\varrho >0\) and \({{\tilde{P}}}(z)=0\) for all other \(z\in R^2{\setminus } \overline{\mathscr {O}}_{{\underline{a}}}\).
We skip the indices N, \(\varepsilon \) and \(\delta \) in what follows and will use (only one of them) in situations when it will be useful to underline the corresponding limit passage.
Indeed, the functional \(L^q((0,T)\times \Omega )\ni \varrho \mapsto R: \int \nolimits _0^T\int \nolimits _{\Omega }(\Lambda \varrho \ln \varrho -\varrho {\mathscr {R}}(\varrho ,s))(t,x)\,\mathrm{d}x\mathrm{d}t\), \(q>1\), is convex and strongly continuous (in particular, it has convex and strongly closed epigraph). Consequently, its epigraph is also weakly closed which is equivalent to its lower weak semi-continuity.
Here we need that \({\underline{a}}>0\) if \(\beta \geqq \gamma +\gamma _{{ BOG}}\).
Estimate (127) is needed only in the borderline case \(\gamma =9/5\) when \(\gamma +\gamma _{{ BOG}}=2\). If \(\gamma >9/5\) then \(\gamma +\gamma _{{ BOG}}>2\) and we can get similar result interpolating \(T_k(\varrho _\delta )-T_k(\varrho )\) between \(L^1\) and \(L^{\gamma +\gamma _{{ BOG}}}\) while using only estimates (126).
If \({\underline{a}}=0\), we need, in addition (142), to make this conclusion.
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Acknowledgements
The work of M. Pokorný was supported by the Czech Science Foundation, Grant No. 16-03230S. A Significant part of the paper was written during the stay of M. Pokorný at the University of Toulon. The authors acknowledge this support.
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Novotný, A., Pokorný, M. Weak Solutions for Some Compressible Multicomponent Fluid Models. Arch Rational Mech Anal 235, 355–403 (2020). https://doi.org/10.1007/s00205-019-01424-2
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DOI: https://doi.org/10.1007/s00205-019-01424-2