# American Institute of Mathematical Sciences

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

## Relative entropies in thermodynamics of complete fluid systems

 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

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.
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.

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##### 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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