A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion

Pierre-Etienne Druet

doi:10.3934/dcdss.2020458

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2021, Volume 14, Issue 11: 4035-4067. Doi: 10.3934/dcdss.2020458 open access

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A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion

Pierre-Etienne Druet

Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany

Received: February 29, 2020

Revised: July 31, 2020

Early access: November 2020

Published: November 2021

The author was supported by the grant D1117/1-1 of the German Science Foundation (DFG)

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Abstract

After the pioneering work by Giovangigli on mathematics of multicomponent flows, several attempts were made to introduce global weak solutions for the PDEs describing the dynamics of fluid mixtures. While the incompressible case with constant density was enlighted well enough due to results by Chen and Jüngel (isothermal case), or Marion and Temam, some open questions remain for the weak solution theory of gas mixtures with their corresponding equations of mixed parabolic–hyperbolic type. For instance, Mucha, Pokorny and Zatorska showed the possibility to stabilise the hyperbolic component by means of the Bresch-Desjardins technique and a regularisation of pressure preventing vacuum. The result by Dreyer, Druet, Gajewski and Guhlke avoids ex machina stabilisations, but the mathematical assumption that the Onsager matrix is uniformly positive on certain subspaces leads, in the dilute limit, to infinite diffusion velocities which are not compatible with the Maxwell-Stefan form of diffusion fluxes. In this paper, we prove the existence of global weak solutions for isothermal and ideal compressible mixtures with natural diffusion. The main new tool is an asymptotic condition imposed at low pressure on the binary Maxwell-Stefan diffusivities, which compensates possibly extreme behaviour of weak solutions in the rarefied regime.

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Mathematics Subject Classification: 35M33, 35Q30, 76N10, 35D30, 35Q35, 35Q79, 76R50.

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References

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