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February  2016, 9(1): 173-183. doi: 10.3934/dcdss.2016.9.173

On weak solutions to a diffuse interface model of a binary mixture of compressible fluids

Eduard Feireisl 1,

1. 

Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1

Received  September 2014 Revised  February 2015 Published  December 2015

We consider the Euler-Cahn-Hilliard system proposed by Lowengrub and Truskinovsky describing the motion of a binary mixture of compressible fluids. We show that the associated initial-value problem possesses infinitely many global-in-time weak solutions for any finite energy initial data. A modification of the method of convex integration is used to prove the result.
Keywords: weak solution, diffuse interface model., Euler-Cahn-Hilliard system.
Mathematics Subject Classification: Primary: 35A01; Secondary: 35Q31, 35Q3.
Citation: Eduard Feireisl. On weak solutions to a diffuse interface model of a binary mixture of compressible fluids. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 173-183. doi: 10.3934/dcdss.2016.9.173
References:
[1]

T. Blesgen, A generalization of the Navier-Stokes equations to two-phase flow, J. Phys. D Appl. Phys., 32 (1999), 1119-1123. doi: 10.1088/0022-3727/32/10/307.  Google Scholar

[2]

E. Chiodaroli, A counterexample to well-posedness of entropy solutions to the compressible Euler system, J. Hyperbolic Differ. Equ., 11 (2014), 493-519. doi: 10.1142/S0219891614500143.  Google Scholar

[3]

E. Chiodaroli, C. De Lellis and O. Kreml, Global ill-posedness of the isentropic system of gas dynamics, Communications on Pure and Applied Mathematics, 68 (2015), 1157-1190. doi: 10.1002/cpa.21537.  Google Scholar

[4]

E. Chiodaroli, E. Feireisl and O. Kreml, On the weak solutions to the equations of a compressible heat conducting gas, Annal. Inst. Poincaré, Anal. Nonlinear., 32 (2015), 225-243. doi: 10.1016/j.anihpc.2013.11.005.  Google Scholar

[5]

C. De Lellis and L. Székelyhidi, Jr., On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 195 (2010), 225-260. doi: 10.1007/s00205-008-0201-x.  Google Scholar

[6]

C. De Lellis and L. Székelyhidi, Jr., The $h$-principle and the equations of fluid dynamics, Bull. Amer. Math. Soc. (N.S.), 49 (2012), 347-375. doi: 10.1090/S0273-0979-2012-01376-9.  Google Scholar

[7]

R. Denk, M. Hieber and J. Prüss, Optimal $L^p-L^q$-estimates for parabolic boundary value problems with inhomogenous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.  Google Scholar

[8]

D. Donatelli, E. Feireisl and P. Marcati, Well/ill posedness for the Euler-Korteweg-Poisson system and related problems, Commun. Partial Differential Equations, 40 (2015), 1314-1335. doi: 10.1080/03605302.2014.972517.  Google Scholar

[9]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 2617-2654. doi: 10.1098/rspa.1998.0273.  Google Scholar

[10]

V. Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal., 3 (1993), 343-401. doi: 10.1007/BF02921318.  Google Scholar

[11]

A. Shnirelman, Weak solutions of incompressible Euler equations, in Handbook of Mathematical Fluid Dynamics, II, North-Holland, Amsterdam, 2003, 87-116. doi: 10.1016/S1874-5792(03)80005-8.  Google Scholar

[12]

L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Anal. and Mech., Heriot-Watt Sympos., Vol. IV, Res. Notes in Math., 39, Pitman, Boston, Mass.-London, 1979, 136-212.  Google Scholar

show all references

References:
[1]

T. Blesgen, A generalization of the Navier-Stokes equations to two-phase flow, J. Phys. D Appl. Phys., 32 (1999), 1119-1123. doi: 10.1088/0022-3727/32/10/307.  Google Scholar

[2]

E. Chiodaroli, A counterexample to well-posedness of entropy solutions to the compressible Euler system, J. Hyperbolic Differ. Equ., 11 (2014), 493-519. doi: 10.1142/S0219891614500143.  Google Scholar

[3]

E. Chiodaroli, C. De Lellis and O. Kreml, Global ill-posedness of the isentropic system of gas dynamics, Communications on Pure and Applied Mathematics, 68 (2015), 1157-1190. doi: 10.1002/cpa.21537.  Google Scholar

[4]

E. Chiodaroli, E. Feireisl and O. Kreml, On the weak solutions to the equations of a compressible heat conducting gas, Annal. Inst. Poincaré, Anal. Nonlinear., 32 (2015), 225-243. doi: 10.1016/j.anihpc.2013.11.005.  Google Scholar

[5]

C. De Lellis and L. Székelyhidi, Jr., On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 195 (2010), 225-260. doi: 10.1007/s00205-008-0201-x.  Google Scholar

[6]

C. De Lellis and L. Székelyhidi, Jr., The $h$-principle and the equations of fluid dynamics, Bull. Amer. Math. Soc. (N.S.), 49 (2012), 347-375. doi: 10.1090/S0273-0979-2012-01376-9.  Google Scholar

[7]

R. Denk, M. Hieber and J. Prüss, Optimal $L^p-L^q$-estimates for parabolic boundary value problems with inhomogenous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.  Google Scholar

[8]

D. Donatelli, E. Feireisl and P. Marcati, Well/ill posedness for the Euler-Korteweg-Poisson system and related problems, Commun. Partial Differential Equations, 40 (2015), 1314-1335. doi: 10.1080/03605302.2014.972517.  Google Scholar

[9]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 2617-2654. doi: 10.1098/rspa.1998.0273.  Google Scholar

[10]

V. Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal., 3 (1993), 343-401. doi: 10.1007/BF02921318.  Google Scholar

[11]

A. Shnirelman, Weak solutions of incompressible Euler equations, in Handbook of Mathematical Fluid Dynamics, II, North-Holland, Amsterdam, 2003, 87-116. doi: 10.1016/S1874-5792(03)80005-8.  Google Scholar

[12]

L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Anal. and Mech., Heriot-Watt Sympos., Vol. IV, Res. Notes in Math., 39, Pitman, Boston, Mass.-London, 1979, 136-212.  Google Scholar

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