Topological Methods in Nonlinear Analysis

Relative entropy method for measure-valued solutions in natural sciences

Tomasz Dębiec, Piotr Gwiazda, Kamila Łyczek, and Agnieszka Świerczewska-Gwiazda

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We describe the applications of the relative entropy framework introduced in [10]. In particular the uniqueness of an entropy solution is proven for a scalar conservation law, using the notion of measure-valued entropy solutions. Further we survey recent results concerning measure-valued-strong uniqueness for a number of physical systems - incompressible and compressible Euler equations, compressible Navier-Stokes, polyconvex elastodynamics and general hyperbolic conservation laws, as well as long-time asymptotics of the McKendrick-Von Foerster equation.

Article information

Topol. Methods Nonlinear Anal., Volume 52, Number 1 (2018), 311-335.

First available in Project Euclid: 18 August 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Dębiec, Tomasz; Gwiazda, Piotr; Łyczek, Kamila; Świerczewska-Gwiazda, Agnieszka. Relative entropy method for measure-valued solutions in natural sciences. Topol. Methods Nonlinear Anal. 52 (2018), no. 1, 311--335. doi:10.12775/TMNA.2018.027.

Export citation


  • J.-J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measure, J. Convex Anal. 4 (1997), 129–148.
  • P. Bella, E. Feireisl and A. Novotný, Dimension reduction for compressible viscous fluids, Acta Appl. Math. 134 (2014), 111–121.
  • Y. Brenier, C. De Lellis and L. Székelyhidi Jr., Weak-strong uniqueness for measure-valued solutions, Comm. Math. Phys. 305(2011), no. 2, 351–361.
  • J. Březina and E. Feireisl, Measure-valued solutions to the complete Euler system, arXiv:1702.04870v1.
  • M. Bulíček, P. Gwiazda, J. Malek and A. Świerczewska-Gwiazda, On scalar hyperbolic laws with discontinuous flux, Math. Models Methods Appl. Sci. 21 (2011), no. 1.
  • M. Bulíček, P. Gwiazda and A. Świerczewska-Gwiazda, On unified theory for scalar conservation laws with fluxes and sources discontinuous with respect to the unknown, J. Differential Equations 262 (2017), 313–364.
  • M. Bulíček, P. Gwiazda and A. Świerczewska-Gwiazda, Multi-dimensional scalar conservation laws with fluxes discontinuous in the unknown and the spatial variable, Math. Models Methods Appl. Sci. 23 (2013), no. 3, 407–439.
  • G-Q. Chen, N. Even and M. Klingenberg, Hyperbolic conservation laws with discontinuous fluxes and hydrodynamic limit for particle systems, J. Differential Equations 245 (2008), 3095–3126.
  • C. Christoforo and A.E. Tzavaras, Relative entropy for hyperbolic-parabolic systems and application to the constitutive theor of thermoviscoelasticity, arXiv:1603.08176.
  • C. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal. 70 (1979), 167–179.
  • C. Dafermos, Hyperbolic conservation laws in continuum physics, third edition, Springer–Verlag, Berlin, 2010.
  • C. De Lellis and L. Székelyhidi Jr., On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal. 195 (2010), no. 1, 225–260.
  • S. Demoulini, D.M.A. Stuart and A.E. Tzavaras, Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics, Arch. Ration. Mech. Anal. 205 (2012), no. 3, 927–961.
  • S. Demoulini, D.M.A. Stuart and A.E. Tzavaras, A variational approximation scheme for three-dimensional elastodynamics with polyconvex energy, Arch. Ration. Mech. Anal. 157 (2001), 325–344.
  • R.J. DiPerna, Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal. 88 (1985), no. 3, 223–270.
  • R.J. DiPerna and A.J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987), no. 4, 667–689.
  • E. Feireisl, B.J. Jin and A. Novotn\' y, Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier–Stokes system, J. Math. Fluid Mech. 14 (2012), 712–730.
  • E. Feireisl, P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Dissipative measure-valued solutions to the compressible Navier–Stokes system, Calc. Var. Partial Differential Equations 55 (2016), no. 6, Art. 141, 20 pp.
  • E. Feireisl and A. Novotný, Weak-strong uniqueness property for the full Navier–Stokes–Fourier system, Arch. Ration. Mech. Anal. 204 (2012), 683–706.
  • U.S. Fjordholm, R. Käppeli, S. Mishra and E. Tadmor, Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws, Found. Comput. Math. 17 (2017), no. 3, 763–827.
  • U.S. Fjordholm, S. Lanthaler and S. Mishra, Statistical solutions of hyperbolic conserwation laws, \romI. Foundations, arXiv:1605.05960.
  • U.S. Fjordholm, S. Mishra and E. Tadmor, On the computation of measure-valued solutions, Acta Numer. 25 (2016), 567–679.
  • J. Giesselmann and A.E. Tzavaras, Stability properties of the Euler–Korteweg system with monotone pressures, Appl. Anal. (2017).
  • P. Gwiazda, On measure-valued solutions to a two-dimensional gravity-driven avalanche flow model, Math. Methods Appl. Sci. 28 (2005), no. 18, 2201–2223.
  • P. Gwiazda, T. Lorenz and A. Marciniak-Czochra A nonlinear structured population model\rom: Lipschitz continuity of measure valued solutions with respect to model ingredients, J. Differential Equations 248 (2010), 2703–2735.
  • P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Weak-strong uniqueness for measure-valued solutions of some compressible fluid models, Nonlinearity 28 (2015), no. 11, 3873–3890.
  • P. Gwiazda, A. Świerczewska-Gwiazda, P. Wittbold and A. Zimmermann, Multi-dimensional scalar balance laws with discontinuous flux, J. Funct. Anal. 267 (2014), no. 8, 2846–2883.
  • P. Gwiazda and E. Wiedemann, Generalized entropy method for the renewal equation with measure data, Commun. Math. Sci. 15 (2017), no. 2, 577–586.
  • D. Kröner and W. Zaj¹czkowski, Measure-valued solutions of the Euler system for ideal compressible polytropic fluids, Math. Methods Appl. Sci. 19 (1996), no. 3, 235–252.
  • S.N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217–243.
  • P.L. Lions, B. Perthame and E. Tadmor, A Kinetic formulation of scalar multidimensional conservation laws, J. Amer. Math. Soc. 7 (1994), 169–191.
  • P. Michel, P. Mischler and B. Perthame, General entropy equations for structured population models and scattering, C.R. Acad. Sci. Paris Ser. I 338 (2–4), 697–702.
  • P. Michel, P. Mischler and B. Perthame, General relative entropy inequality: an illustration on growth models, J. Math. Pures Appl. 84 (2005), 1235–1260.
  • S. Mischler, B. Perthame and L. Ryzhik, Stability in a nonlinear population maturation model, Math. Models Methods Appl. Sci. 12 (2002), 1751–1772; Lecture Notes in Math. 1048, 60–110.
  • J. Nečas, J. Málek, M. Rokyta and M. R\r užička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman and Hall/CRC, 1996.
  • J. Neustupa, Measure-valued solutions of the Euler and Navier–Stokes equations for compressible barotropic fluids, Math. Nachr. 163:217–227, 1993.
  • B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser, Basel, 2007.
  • B. Perthame and E. Tadmor, A kinetic equation with kinetic entropy functions for scalar conservation laws, Commun. Math. Phys. 136 (1991), 501–517.
  • B. Perthame and A.E. Tzavaras, Kinetic formulation for systems of two conservation laws and elastodynamics, Arch. Ration. Mech. Anal. 155 (2000), no. 1, 1–48.
  • L. Székelyhidi Jr. and E. Wiedemann, Young measures generated by ideal incompressible fluid flows, Arch. Rational Mech. Anal. 206 (2012), no. 1, 333–366.
  • A. Szepessy, An existence result for scalar conservation laws using measure valued solutions, Comm. Partial Differential Equations 14 (1989), no. 10, 1329–1350.
  • L. Tartar, Compensated compactness method applied to systems of conservation laws, Systems Nonlinear PDE (J.M. Ball, ed.), NATO ASI Series, C. Reidel Publishing Col., 1983.
  • E. Wiedemann, Weak-strong uniqueness in fluid dynamics, Partial Differential Equations in Fluid Mechanics (C.L. Fefferman, J.C. Robinson and J.L. Rodrigo, eds.), Cambridge University Press.