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

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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

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

Abstract
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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. 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. 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. 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, 32 (2015), 225. 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. 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. 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. 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. 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. 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. doi: 10.1007/BF02921318. Google Scholar
[11]	A. Shnirelman, Weak solutions of incompressible Euler equations,, in Handbook of Mathematical Fluid Dynamics, (2003), 87. 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., (1979), 136. Google Scholar

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References:

[1]	T. Blesgen, A generalization of the Navier-Stokes equations to two-phase flow,, J. Phys. D Appl. Phys., 32 (1999), 1119. 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. 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. 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, 32 (2015), 225. 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. 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. 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. 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. 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. 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. doi: 10.1007/BF02921318. Google Scholar
[11]	A. Shnirelman, Weak solutions of incompressible Euler equations,, in Handbook of Mathematical Fluid Dynamics, (2003), 87. 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., (1979), 136. Google Scholar

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