November  2019, 39(11): 6175-6206. doi: 10.3934/dcds.2019269

Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity

1. 

Department of Mathematics and Statistics, University of Cyprus, Nicosia 1678, Cyprus

2. 

Computer, Electrical, Mathematical Sciences & Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia

* Corresponding author: Christoforou was partially supported by the Internal grant SBLawsMechGeom #21036 from University of Cyprus

Received  August 2018 Revised  November 2018 Published  August 2019

Fund Project: This project has received funding from the European Union's Horizon 2020 programme under the Marie Sklodowska-Curie grant agreement No 642768

For the system of polyconvex adiabatic thermoelasticity, we define a notion of dissipative measure-valued solution, which can be considered as the limit of a viscosity approximation. We embed the system into a symmetrizable hyperbolic one in order to derive the relative entropy. Exploiting the weak-stability properties of the transport and stretching identities, we base our analysis in the original variables, instead of the symmetric ones (in which the entropy is convex) and we prove measure-valued weak versus strong uniqueness using the averaged relative entropy inequality.

Citation: Cleopatra Christoforou, Myrto Galanopoulou, Athanasios E. Tzavaras. Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6175-6206. doi: 10.3934/dcds.2019269
References:
[1]

J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures, J. Convex Analysis, 4 (1997), 129-147.   Google Scholar

[2]

J. M. Ball, A version of the fundamental theorem for Young measures, PDEs and Continuum Models of Phase Transitions (Nice, 1988), 334 (2005), 207-215.  doi: 10.1007/BFb0024945.  Google Scholar

[3]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63 (1976/77), 337-403.  doi: 10.1007/BF00279992.  Google Scholar

[4]

J. M. Ball, Some open problems in elasticity, in Geometry, Mechanics, and Dynamics, Springer, New York, (2002), 3–59. doi: 10.1007/0-387-21791-6_1.  Google Scholar

[5]

Y. BrenierC. De Lellis and L. Székelyhidi Jr., Weak-strong uniqueness for measure-valued solutions, Comm. Math. Physics, 305 (2011), 351-361.  doi: 10.1007/s00220-011-1267-0.  Google Scholar

[6]

J. Březina and E. Feireisl, Measure-valued solutions to the complete Euler system, J. Math. Soc. Japan, 70 (2018), 1227–1245, arXiv: 1702.04878. doi: 10.2969/jmsj/77337733.  Google Scholar

[7]

H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, Wiley, New York, 1985. Google Scholar

[8]

C. ChristoforouM. Galanopoulou and A. E. Tzavaras, A symmetrizable extension of polyconvex thermoelasticity and applications to zero-viscosity limits and weak-strong uniqueness, Comm. Partial Differential Equations, 43 (2018), 1019-1050.  doi: 10.1080/03605302.2018.1456551.  Google Scholar

[9]

C. Christoforou and A. E. Tzavaras, Relative entropy for hyperbolic–parabolic systems and application to the constitutive theory of thermoviscoelasticity, Arch. Rational Mech. Anal., 229 (2018), 1-52.  doi: 10.1007/s00205-017-1212-2.  Google Scholar

[10]

B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal., 13 (1963), 167-178.  doi: 10.1007/BF01262690.  Google Scholar

[11]

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179.  doi: 10.1007/BF00250353.  Google Scholar

[12]

C. M. Dafermos, Stability of motions of thermoelastic fluids, Journal of Thermal Stresses, 2 (1979), 127-134.  doi: 10.1080/01495737908962394.  Google Scholar

[13]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325. Springer-Verlag, Berlin, 2005. doi: 10.1007/3-540-29089-3.  Google Scholar

[14]

S. DemouliniD. 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.  doi: 10.1007/s002050100137.  Google Scholar

[15]

S. DemouliniD. M. A. Stuart and A. E. Tzavaras, Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics, Arch. Ration. Mech. Anal., 205 (2012), 927-961.  doi: 10.1007/s00205-012-0523-6.  Google Scholar

[16]

R. J. DiPerna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J., 28 (1979), 137-188.  doi: 10.1512/iumj.1979.28.28011.  Google Scholar

[17]

R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270.  doi: 10.1007/BF00752112.  Google Scholar

[18]

R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Commun. Math. Phys., 108 (1987), 667-689.  doi: 10.1007/BF01214424.  Google Scholar

[19]

L. C. Evans, Entropy and Partial Differential Equations, UC Berkely, Lecture Notes, Available through: https://math.berkeley.edu/ evans/entropy.and.PDE.pdf. Google Scholar

[20]

U. S. FjordholmR. KäppeliS. Mishra and E. Tadmor, Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws, Found. Comput. Math., 17 (2017), 763-827.  doi: 10.1007/s10208-015-9299-z.  Google Scholar

[21]

K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. U.S.A., 68 (1971), 1686-1688.  doi: 10.1073/pnas.68.8.1686.  Google Scholar

[22]

P. GwiazdaA. Świerczewska-Gwiazda and E. Wiedemann, Weak-strong uniqueness for measure-valued solutions of some compressible fluid models, Nonlinearity, 28 (2015), 3873-3890.  doi: 10.1088/0951-7715/28/11/3873.  Google Scholar

[23]

T. Qin, Symmetrizing nonlinear elastodynamic system, J. Elasticity, 50 (1998), 245-252.  doi: 10.1023/A:1007488013851.  Google Scholar

[24]

L. Tartar, The compensated compactness method applied to systems of conservation laws, C. Reidel Publishing Col., 111 (1983), 263-285.   Google Scholar

[25]

C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Second edition. Springer Verlag, Berlin, 1992. doi: 10.1007/978-3-662-13183-1.  Google Scholar

show all references

References:
[1]

J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures, J. Convex Analysis, 4 (1997), 129-147.   Google Scholar

[2]

J. M. Ball, A version of the fundamental theorem for Young measures, PDEs and Continuum Models of Phase Transitions (Nice, 1988), 334 (2005), 207-215.  doi: 10.1007/BFb0024945.  Google Scholar

[3]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63 (1976/77), 337-403.  doi: 10.1007/BF00279992.  Google Scholar

[4]

J. M. Ball, Some open problems in elasticity, in Geometry, Mechanics, and Dynamics, Springer, New York, (2002), 3–59. doi: 10.1007/0-387-21791-6_1.  Google Scholar

[5]

Y. BrenierC. De Lellis and L. Székelyhidi Jr., Weak-strong uniqueness for measure-valued solutions, Comm. Math. Physics, 305 (2011), 351-361.  doi: 10.1007/s00220-011-1267-0.  Google Scholar

[6]

J. Březina and E. Feireisl, Measure-valued solutions to the complete Euler system, J. Math. Soc. Japan, 70 (2018), 1227–1245, arXiv: 1702.04878. doi: 10.2969/jmsj/77337733.  Google Scholar

[7]

H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, Wiley, New York, 1985. Google Scholar

[8]

C. ChristoforouM. Galanopoulou and A. E. Tzavaras, A symmetrizable extension of polyconvex thermoelasticity and applications to zero-viscosity limits and weak-strong uniqueness, Comm. Partial Differential Equations, 43 (2018), 1019-1050.  doi: 10.1080/03605302.2018.1456551.  Google Scholar

[9]

C. Christoforou and A. E. Tzavaras, Relative entropy for hyperbolic–parabolic systems and application to the constitutive theory of thermoviscoelasticity, Arch. Rational Mech. Anal., 229 (2018), 1-52.  doi: 10.1007/s00205-017-1212-2.  Google Scholar

[10]

B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal., 13 (1963), 167-178.  doi: 10.1007/BF01262690.  Google Scholar

[11]

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179.  doi: 10.1007/BF00250353.  Google Scholar

[12]

C. M. Dafermos, Stability of motions of thermoelastic fluids, Journal of Thermal Stresses, 2 (1979), 127-134.  doi: 10.1080/01495737908962394.  Google Scholar

[13]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325. Springer-Verlag, Berlin, 2005. doi: 10.1007/3-540-29089-3.  Google Scholar

[14]

S. DemouliniD. 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.  doi: 10.1007/s002050100137.  Google Scholar

[15]

S. DemouliniD. M. A. Stuart and A. E. Tzavaras, Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics, Arch. Ration. Mech. Anal., 205 (2012), 927-961.  doi: 10.1007/s00205-012-0523-6.  Google Scholar

[16]

R. J. DiPerna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J., 28 (1979), 137-188.  doi: 10.1512/iumj.1979.28.28011.  Google Scholar

[17]

R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270.  doi: 10.1007/BF00752112.  Google Scholar

[18]

R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Commun. Math. Phys., 108 (1987), 667-689.  doi: 10.1007/BF01214424.  Google Scholar

[19]

L. C. Evans, Entropy and Partial Differential Equations, UC Berkely, Lecture Notes, Available through: https://math.berkeley.edu/ evans/entropy.and.PDE.pdf. Google Scholar

[20]

U. S. FjordholmR. KäppeliS. Mishra and E. Tadmor, Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws, Found. Comput. Math., 17 (2017), 763-827.  doi: 10.1007/s10208-015-9299-z.  Google Scholar

[21]

K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. U.S.A., 68 (1971), 1686-1688.  doi: 10.1073/pnas.68.8.1686.  Google Scholar

[22]

P. GwiazdaA. Świerczewska-Gwiazda and E. Wiedemann, Weak-strong uniqueness for measure-valued solutions of some compressible fluid models, Nonlinearity, 28 (2015), 3873-3890.  doi: 10.1088/0951-7715/28/11/3873.  Google Scholar

[23]

T. Qin, Symmetrizing nonlinear elastodynamic system, J. Elasticity, 50 (1998), 245-252.  doi: 10.1023/A:1007488013851.  Google Scholar

[24]

L. Tartar, The compensated compactness method applied to systems of conservation laws, C. Reidel Publishing Col., 111 (1983), 263-285.   Google Scholar

[25]

C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Second edition. Springer Verlag, Berlin, 1992. doi: 10.1007/978-3-662-13183-1.  Google Scholar

[1]

Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032

[2]

Azmy S. Ackleh, Vinodh K. Chellamuthu, Kazufumi Ito. Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 233-258. doi: 10.3934/mbe.2015.12.233

[3]

Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020041

[4]

Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017

[5]

Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056

[6]

Simona Fornaro, Stefano Lisini, Giuseppe Savaré, Giuseppe Toscani. Measure valued solutions of sub-linear diffusion equations with a drift term. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1675-1707. doi: 10.3934/dcds.2012.32.1675

[7]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781

[8]

Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081

[9]

Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun. Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks & Heterogeneous Media, 2013, 8 (4) : 943-968. doi: 10.3934/nhm.2013.8.943

[10]

Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288

[11]

Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192

[12]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461

[13]

Erik M. Bollt, Joseph D. Skufca, Stephen J . McGregor. Control entropy: A complexity measure for nonstationary signals. Mathematical Biosciences & Engineering, 2009, 6 (1) : 1-25. doi: 10.3934/mbe.2009.6.1

[14]

Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159

[15]

Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic & Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701

[16]

Denis Serre, Alexis F. Vasseur. The relative entropy method for the stability of intermediate shock waves; the rich case. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4569-4577. doi: 10.3934/dcds.2016.36.4569

[17]

María Anguiano, Tomás Caraballo, José Real, José Valero. Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 307-326. doi: 10.3934/dcdsb.2010.14.307

[18]

Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673

[19]

Jisang Yoo. Decomposition of infinite-to-one factor codes and uniqueness of relative equilibrium states. Journal of Modern Dynamics, 2018, 13: 271-284. doi: 10.3934/jmd.2018021

[20]

Ramon Quintanilla. Structural stability and continuous dependence of solutions of thermoelasticity of type III. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 463-470. doi: 10.3934/dcdsb.2001.1.463

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (74)
  • HTML views (93)
  • Cited by (0)

[Back to Top]