Relative entropies in thermodynamics of complete fluid systems

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September 2012, 32(9): 3059-3080. doi: 10.3934/dcds.2012.32.3059

Relative entropies in thermodynamics of complete fluid systems

Eduard Feireisl ^1,

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

Received November 2011 Revised March 2012 Published April 2012

Abstract
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We introduce the notion of relative entropy in the framework of thermodynamics of compressible, viscous and heat conducting fluids. The relative entropy is constructed on the basis of a thermodynamic potential called ballistic free energy and provides stability of solutions to the associated Navier-Stokes-Fourier system with respect to perturbations. The theory is illustrated by applications to problems related to the long time behavior of solutions and the problem of weak-strong uniqueness.

Keywords: relative entropy, qualitative properties of compressible fluids., Navier-Stokes-Fourier system.

Mathematics Subject Classification: Primary: 35Q30; Secondary: 35B2.

Citation: Eduard Feireisl. Relative entropies in thermodynamics of complete fluid systems. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3059-3080. doi: 10.3934/dcds.2012.32.3059

References:

[1]	S. E. Bechtel, F. J. Rooney and M. G. Forest, Connection between stability, convexity of internal energy, and the second law for compressible Newtonian fuids, J. Appl. Mech., 72 (2005), 299-300. doi: 10.1115/1.1831297.
[2]	E. Becker, "Gasdynamik," (German), Leitfäden der Angewandten Mathematik und Mechanik, Band 6, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1966.
[3]	D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl. (9), 87 (2007), 57-90. doi: 10.1016/j.matpur.2006.11.001.
[4]	J. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatshefte Math., 133 (2001), 1-82. doi: 10.1007/s006050170032.
[5]	C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179. doi: 10.1007/BF00250353.
[6]	B. Desjardins, Regularity of weak solutions of the compressible isentropic Navier-Stokes equations, Commun. Partial Differential Equations, 22 (1997), 977-1008.
[7]	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.
[8]	S. Eliezer, A. Ghatak and H. Hora, "An Introduction to Equations of States, Theory and Applications," Cambridge University Press, Cambridge, 1986.
[9]	E. Feireisl and Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system, Arch. Rational Mech. Anal., to appear, 2012.
[10]	E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids," Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009.
[11]	E. Feireisl, A. Novotný and B. J. Jin, Relative entropies, suitable weak solutions, and uniqueness for the compressible Navier-Stokes system, J. Math. Fluid Mechanics, to appear, 2012.
[12]	E. Feireisl, A. Novotný and Y. Sun, Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids, Indiana Univ. Math. J., to appear, 2012.
[13]	E. Feireisl and D. Pražák, "Asymptotic Behavior of Dynamical Systems in Fluid Mechanics," AIMS Series on Applied Mathematics, 4, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2010.
[14]	P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech., published online, 2010.
[15]	A. Mellet and A. Vasseur, Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations,, SIAM J. Math. Anal., 39 (): 1344. doi: 10.1137/060658199.
[16]	L. Saint-Raymond, Hydrodynamic limits: Some improvements of the relative entropy method, Annal. I. H. Poincaré Anal. Non Linéaire, 26 (2009), 705-744.

show all references

References:

[1]	S. E. Bechtel, F. J. Rooney and M. G. Forest, Connection between stability, convexity of internal energy, and the second law for compressible Newtonian fuids, J. Appl. Mech., 72 (2005), 299-300. doi: 10.1115/1.1831297.
[2]	E. Becker, "Gasdynamik," (German), Leitfäden der Angewandten Mathematik und Mechanik, Band 6, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1966.
[3]	D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl. (9), 87 (2007), 57-90. doi: 10.1016/j.matpur.2006.11.001.
[4]	J. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatshefte Math., 133 (2001), 1-82. doi: 10.1007/s006050170032.
[5]	C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179. doi: 10.1007/BF00250353.
[6]	B. Desjardins, Regularity of weak solutions of the compressible isentropic Navier-Stokes equations, Commun. Partial Differential Equations, 22 (1997), 977-1008.
[7]	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.
[8]	S. Eliezer, A. Ghatak and H. Hora, "An Introduction to Equations of States, Theory and Applications," Cambridge University Press, Cambridge, 1986.
[9]	E. Feireisl and Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system, Arch. Rational Mech. Anal., to appear, 2012.
[10]	E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids," Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009.
[11]	E. Feireisl, A. Novotný and B. J. Jin, Relative entropies, suitable weak solutions, and uniqueness for the compressible Navier-Stokes system, J. Math. Fluid Mechanics, to appear, 2012.
[12]	E. Feireisl, A. Novotný and Y. Sun, Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids, Indiana Univ. Math. J., to appear, 2012.
[13]	E. Feireisl and D. Pražák, "Asymptotic Behavior of Dynamical Systems in Fluid Mechanics," AIMS Series on Applied Mathematics, 4, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2010.
[14]	P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech., published online, 2010.
[15]	A. Mellet and A. Vasseur, Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations,, SIAM J. Math. Anal., 39 (): 1344. doi: 10.1137/060658199.
[16]	L. Saint-Raymond, Hydrodynamic limits: Some improvements of the relative entropy method, Annal. I. H. Poincaré Anal. Non Linéaire, 26 (2009), 705-744.

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