Artificial Compressibility Method for the Navier–Stokes–Maxwell–Stefan System


The Navier–Stokes–Maxwell–Stefan system describes the dynamics of an incompressible gaseous mixture in isothermal condition. In this paper we set up an artificial compressibility type approximation. In particular we focus on the existence of solution for the approximated system and the convergence to the incompressible case. The existence of the approximating system is proved by means of semidiscretization in time and by estimating the fractional time derivative.

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    To choose properly the constant C, observe that if we impose to zero the right hand side of (4.10), for any fixed \(\varepsilon >0\) we have an equation like \(x(x-a)+\varepsilon y(y-b)=0\), with \(a,b>0\) given. But this equation represent an ellipse, and so the condition in (4.11) is reduced to find a constant \(C>0\), such that the straight line \(x+y=C\), for \(x,y>0\), lies entirely in the exterior of the ellipse, and this is possible since the ellipse is bounded.


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Correspondence to Donatella Donatelli.

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Dolce, M., Donatelli, D. Artificial Compressibility Method for the Navier–Stokes–Maxwell–Stefan System. J Dyn Diff Equat 33, 35–62 (2021).

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  • Stefan–Maxwell Navier Stokes equation
  • Artificial compressibility method
  • Fixed point methods