Weak solutions for a bi-fluid model for a mixture of two compressible non interacting fluids
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We investigate a version of one velocity Baer-Nunziato model with dissipation for the mixture of two compressible fluids with the goal to prove for it the existence of weak solutions for arbitrary large initial data on a large time interval. We transform the one velocity Baer-Nunziato system to another "more academic" system which possesses the clear "Navier-Stokes structure". We solve the new system by adapting to its structure the Lions approach for solving the (mono-fluid) compressible Navier-Stokes equations. An extension of the theory of renormalized solutions to the transport equation to more continuity equations with renormalizing functions of several variables is essential in this process. We derive a criterion of almost uniqueness for the renormalized solutions to the pure transport equation without the classical assumption on the boundedness of the divergence of the transporting velocity. This result does not follow from the DiPerna-Lions transport theory and it is of independent interest. This criterion plays the crucial role in the identification of the weak solutions to the original one velocity Baer-Nunziato problem starting from the weak solutions of the academic problem. As far as we know, this is the first result on the existence of weak solutions for a version of the one velocity bi-fluid system of the Baer-Nunziato type in the mathematical literature.
Keywordsbi-fluid system multifluid system Baer-Nunziato system, compressible Navier-Stokes equations transport equation continuity equation large data weak solution
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- 3.Ambrosio L, Crippa G. Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields. In: Transport Equations and Multi-D Hyperbolic Conservation Laws. Lecture Notes of the Unione Matematica Italiana, vol. 5. Berlin-Heidelberg: Springer, 3–57zbMATHGoogle Scholar
- 5.Bianchini S, Bonicatto P. A uniqueness result for the decomposition of vector fields in Rd. https://people.sissa.it/bianchin/Papers/Linear_Transport/weak_Bressan_conj_master.pdfGoogle Scholar
- 6.Bresch D, Desjardins B, Ghidaglia J-M, et al. Multifluid models including compressible fluids. In: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Cham: Springer, 2018, 1–52Google Scholar
- 9.Dallet S. A comparative study of numerical schemes for the Baer-Nunziato model. Int J Finite Vol, 2016, 13: 1–37Google Scholar
- 15.Guillemaud V. Modélisation et simulation numérique des écoulements diphasiques par une approche bifuide à deux pressions. PhD Thesis. Marseille: Université de Provence-Aix-Marseille I, 2007Google Scholar
- 17.Lions P-L. Mathematical Topics in Fluid Mechanics: Compressible Models, Volume 2. Oxford Lecture Series in Mathematics and Its Applications, vol. 10. Oxford: Oxford Science Publications, 1998Google Scholar
- 19.Novotný A, Pokorny M. Weak solutions for some compressible multicomponent fluid models. Arch Ration Mech Anal, 2019, in pressGoogle Scholar
- 20.Novotný A, Straškraba I. Introduction to the Mathematical Theory of Compressible Flow. Oxford Lecture Series in Mathematics and Its Applications, vol. 27. Oxford: Oxford University Press, 2004Google Scholar
- 22.Wen H. Global existence of weak solution to compressible two-fluid model without any domination condition in three dimensions. ArXiv:1902.05190, 2019Google Scholar