# American Institute of Mathematical Sciences

August  2022, 15(8): 2215-2232. doi: 10.3934/dcdss.2022091

## Two phase flows of compressible viscous fluids

 1 Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, CZ-115 67 Praha 1, Czech Republic 2 IMATH, EA 2134, Université de Toulon, BP 20132, 83957 La Garde, France

* Corresponding author: Eduard Feireisl

Received  July 2021 Revised  March 2022 Published  August 2022 Early access  April 2022

Fund Project: The work of E.F. was supported by the Czech Sciences Foundation (GAČR), Grant Agreement 21–02411S.

We introduce a new concept of dissipative varifold solution to models of two phase compressible viscous fluids. In contrast with the existing approach based on the Young measure description, the new formulation is variational combining the energy and momentum balance in a single inequality. We show the existence of dissipative varifold solutions for a large class of general viscous fluids with non–linear dependence of the viscous stress on the symmetric velocity gradient.

Citation: Eduard Feireisl, Antonín Novotný. Two phase flows of compressible viscous fluids. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2215-2232. doi: 10.3934/dcdss.2022091
##### References:
 [1] A. Abbatiello, E. Feireisl and A. Novotný, Generalized solutions to models of compressible viscous fluids, Discrete Contin. Dyn. Syst., 41 (2021), 1-28.  doi: 10.3934/dcds.2020345. [2] H. Abels, On generalized solutions of two-phase flows for viscous incompressible fluids, Interfaces Free Bound., bf 9 (2007), 31–65. doi: 10.4171/IFB/155. [3] H. Abels, On the notion of generalized solutions of viscous incompressible two-phase flows, In Kyoto Conference on the Navier-Stokes Equations and their Applications, RIMS Kôkyûroku Bessatsu, B1, pages 1–19. Res. Inst. Math. Sci. (RIMS), Kyoto, 2007. [4] D. M. Ambrose, M. C. Lopes Filho, H. J. Nussenzveig Lopes and W. A. Strauss, Transport of interfaces with surface tension by 2D viscous flows, Interfaces Free Bound., 12 (2010), 23-44.  doi: 10.4171/IFB/225. [5] H. A. Barnes, Shear-Thickening ("Dilatancy") in suspensions on nonaggregating solidparticles dispersed in Newtonian liquids, J. Rheology, 33 (1989), 329-366. [6] M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda, On unsteady flows of implicitly constituted incompressible fluids, SIAM J. Math. Anal., 44 (2012), 2756-2801.  doi: 10.1137/110830289. [7] I. Denisova and V. A. Solonnikov, Local and Global Solvability of Free Boundary Value Problems Near Equalibria, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Vol. 2 eds. Y. Giga, A. Novotny, Springer, 2018. doi: 10.1007/978-3-319-13344-7. [8] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835. [9] E. Feireisl, X. Liao and J. Málek, Global weak solutions to a class of non-Newtonian compressible fluids, Math. Meth. Appl. Sci., 38 (2015), 3482-3494.  doi: 10.1002/mma.3432. [10] J. Fischer and S. Hensel, Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension, Arch. Rational Mech. Anal., 236 (2020), 967-1087.  doi: 10.1007/s00205-019-01486-2. [11] P. I. Plotnikov, Generalized solutions of a problem on the motion of a non-Newtonian fluid with a free boundary, Sibirsk. Mat. Zh., 34 (1993), 127–141, iii, ix. doi: 10.1007/BF00975173. [12] P. I. Plotnikov, Varifold solutions of a free boundary problem in viscous fluid dynamics, In Free Boundary Problems in Fluid Flow with Applications (Montreal, PQ, 1990), volume 282 of Pitman Res. Notes Math. Ser., pages 28–32. Longman Sci. Tech., Harlow, 1993. [13] P. I. Plotnikov, Compressible Stokes flow driven by capillarity on a free surface, In Navier-Stokes Equations and Related Nonlinear Problems (Palanga, 1997), pages 217–238. VSP, Utrecht, 1998. [14] J.-F. Rodrigues, On the mathematical analysis of thick fluids, J. Math. Sci. (N.Y.), 210 (2015), 835-848.  doi: 10.1007/s10958-015-2594-z.

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##### References:
 [1] A. Abbatiello, E. Feireisl and A. Novotný, Generalized solutions to models of compressible viscous fluids, Discrete Contin. Dyn. Syst., 41 (2021), 1-28.  doi: 10.3934/dcds.2020345. [2] H. Abels, On generalized solutions of two-phase flows for viscous incompressible fluids, Interfaces Free Bound., bf 9 (2007), 31–65. doi: 10.4171/IFB/155. [3] H. Abels, On the notion of generalized solutions of viscous incompressible two-phase flows, In Kyoto Conference on the Navier-Stokes Equations and their Applications, RIMS Kôkyûroku Bessatsu, B1, pages 1–19. Res. Inst. Math. Sci. (RIMS), Kyoto, 2007. [4] D. M. Ambrose, M. C. Lopes Filho, H. J. Nussenzveig Lopes and W. A. Strauss, Transport of interfaces with surface tension by 2D viscous flows, Interfaces Free Bound., 12 (2010), 23-44.  doi: 10.4171/IFB/225. [5] H. A. Barnes, Shear-Thickening ("Dilatancy") in suspensions on nonaggregating solidparticles dispersed in Newtonian liquids, J. Rheology, 33 (1989), 329-366. [6] M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda, On unsteady flows of implicitly constituted incompressible fluids, SIAM J. Math. Anal., 44 (2012), 2756-2801.  doi: 10.1137/110830289. [7] I. Denisova and V. A. Solonnikov, Local and Global Solvability of Free Boundary Value Problems Near Equalibria, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Vol. 2 eds. Y. Giga, A. Novotny, Springer, 2018. doi: 10.1007/978-3-319-13344-7. [8] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835. [9] E. Feireisl, X. Liao and J. Málek, Global weak solutions to a class of non-Newtonian compressible fluids, Math. Meth. Appl. Sci., 38 (2015), 3482-3494.  doi: 10.1002/mma.3432. [10] J. Fischer and S. Hensel, Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension, Arch. Rational Mech. Anal., 236 (2020), 967-1087.  doi: 10.1007/s00205-019-01486-2. [11] P. I. Plotnikov, Generalized solutions of a problem on the motion of a non-Newtonian fluid with a free boundary, Sibirsk. Mat. Zh., 34 (1993), 127–141, iii, ix. doi: 10.1007/BF00975173. [12] P. I. Plotnikov, Varifold solutions of a free boundary problem in viscous fluid dynamics, In Free Boundary Problems in Fluid Flow with Applications (Montreal, PQ, 1990), volume 282 of Pitman Res. Notes Math. Ser., pages 28–32. Longman Sci. Tech., Harlow, 1993. [13] P. I. Plotnikov, Compressible Stokes flow driven by capillarity on a free surface, In Navier-Stokes Equations and Related Nonlinear Problems (Palanga, 1997), pages 217–238. VSP, Utrecht, 1998. [14] J.-F. Rodrigues, On the mathematical analysis of thick fluids, J. Math. Sci. (N.Y.), 210 (2015), 835-848.  doi: 10.1007/s10958-015-2594-z.
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