The relative entropy method for inhomogeneous systems of balance laws

Christoforou, Cleopatra

doi:10.1090/qam/1577

Author: Cleopatra Christoforou
Journal: Quart. Appl. Math. 79 (2021), 201-227
MSC (2010): Primary 35L65, 35A02, 35B35; Secondary 35Q74, 35Q35, 35L45, 35K45
DOI: https://doi.org/10.1090/qam/1577
Published electronically: August 18, 2020
MathSciNet review: 4246491
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: General hyperbolic systems of balance laws with inhomogeneity in space and time in all constitutive functions are studied in the context of relative entropy. A framework is developed in this setting that contributes to a measure-valued weak vs. strong uniqueness theorem, a stability theorem of viscous solutions and a convergence theorem as the viscosity parameter tends to zero. The main goal of this paper is to develop hypotheses under which the relative entropy framework can still be applied. Examples of systems with inhomogeneity that have different characteristics are presented and the hypotheses are discussed in the setting of each example.

References

J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures, J. Convex Anal. 4 (1997), no. 1, 129–147. MR 1459885
D. Amadori and G. Guerra, Global weak solutions for systems of balance laws, Appl. Math. Lett. 12 (1999), no. 6, 123–127. MR 1751419, DOI https://doi.org/10.1016/S0893-9659%2899%2900090-7
J. M. Ball, A version of the fundamental theorem for Young measures, PDEs and continuum models of phase transitions (Nice, 1988) Lecture Notes in Phys., vol. 344, Springer, Berlin, 1989, pp. 207–215. MR 1036070, DOI https://doi.org/10.1007/BFb0024945
Yann Brenier, Camillo De Lellis, and László Székelyhidi Jr., Weak-strong uniqueness for measure-valued solutions, Comm. Math. Phys. 305 (2011), no. 2, 351–361. MR 2805464, DOI https://doi.org/10.1007/s00220-011-1267-0
Gui-Qiang Chen and Cleopatra Christoforou, Solutions for a nonlocal conservation law with fading memory, Proc. Amer. Math. Soc. 135 (2007), no. 12, 3905–3915. MR 2341940, DOI https://doi.org/10.1090/S0002-9939-07-08942-3
Cleopatra C. Christoforou, Hyperbolic systems of balance laws via vanishing viscosity, J. Differential Equations 221 (2006), no. 2, 470–541. MR 2196486, DOI https://doi.org/10.1016/j.jde.2005.03.010
Cleopatra Christoforou, Systems of hyperbolic conservation laws with memory, J. Hyperbolic Differ. Equ. 4 (2007), no. 3, 435–478. MR 2339804, DOI https://doi.org/10.1142/S0219891607001215

C. Christoforou, Isometric immersions via continum mechanics, Chapter in Partial Differential Equations: Ambitious Mathematics for Real-Life Applications, D. Donatelli and C. Simeoni (eds.), SEMA SIMAI Springer Series, Springer, submitted.

Cleopatra Christoforou and Laura V. Spinolo, A uniqueness criterion for viscous limits of boundary Riemann problems, J. Hyperbolic Differ. Equ. 8 (2011), no. 3, 507–544. MR 2831272, DOI https://doi.org/10.1142/S0219891611002482

Cleopatra Christoforou and Laura V. Spinolo, Boundary layers for self-similar viscous approximations of nonlinear hyperbolic systems, Quart. Appl. Math. 71 (2013), no. 3, 433–453. MR 3112822, DOI https://doi.org/10.1090/S0033-569X-2013-01284-6

Cleopatra Christoforou and Athanasios E. Tzavaras, Relative entropy for hyperbolic-parabolic systems and application to the constitutive theory of thermoviscoelasticity, Arch. Ration. Mech. Anal. 229 (2018), no. 1, 1–52. MR 3799089, DOI https://doi.org/10.1007/s00205-017-1212-2

Kyudong Choi and Alexis F. Vasseur, Short-time stability of scalar viscous shocks in the inviscid limit by the relative entropy method, SIAM J. Math. Anal. 47 (2015), no. 2, 1405–1418. MR 3333670, DOI https://doi.org/10.1137/140961523

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal. 70 (1979), no. 2, 167–179. MR 546634, DOI https://doi.org/10.1007/BF00250353

C. M. Dafermos, Stability of motions of thermoelastic fluids, J. Thermal Stresses 2 (1979), no. 1, 127–134.

C. M. Dafermos, Development of singularities in the motion of materials with fading memory, Arch. Rational Mech. Anal. 91 (1985), no. 3, 193–205. MR 806001, DOI https://doi.org/10.1007/BF00250741

Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 4th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2016. MR 3468916

Constantine M. Dafermos, Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Arch. Rational Mech. Anal. 52 (1973), 1–9. MR 340837, DOI https://doi.org/10.1007/BF00249087

C. M. Dafermos and L. Hsiao, Hyperbolic systems and balance laws with inhomogeneity and dissipation, Indiana Univ. Math. J. 31 (1982), no. 4, 471–491. MR 662914, DOI https://doi.org/10.1512/iumj.1982.31.31039

C. M. Dafermos, Solutions in $L^\infty $ for a conservation law with memory, Analyse mathématique et applications, Gauthier-Villars, Montrouge, 1988, pp. 117–128. MR 956955

Sophia Demoulini, David M. A. Stuart, and Athanasios E. Tzavaras, Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics, Arch. Ration. Mech. Anal. 205 (2012), no. 3, 927–961. MR 2960036, DOI https://doi.org/10.1007/s00205-012-0523-6

Ronald J. DiPerna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J. 28 (1979), no. 1, 137–188. MR 523630, DOI https://doi.org/10.1512/iumj.1979.28.28011

Ronald J. DiPerna and Andrew J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987), no. 4, 667–689. MR 877643

Eduard Feireisl and Antonín Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal. 204 (2012), no. 2, 683–706. MR 2909912, DOI https://doi.org/10.1007/s00205-011-0490-3

Ulrik S. Fjordholm, Roger Käppeli, Siddhartha Mishra, and Eitan Tadmor, Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws, Found. Comput. Math. 17 (2017), no. 3, 763–827. MR 3648106, DOI https://doi.org/10.1007/s10208-015-9299-z

Jan Giesselmann and Athanasios E. Tzavaras, Singular limiting induced from continuum solutions and the problem of dynamic cavitation, Arch. Ration. Mech. Anal. 212 (2014), no. 1, 241–281. MR 3162478, DOI https://doi.org/10.1007/s00205-013-0677-x

Piotr Gwiazda, Agnieszka Świerczewska-Gwiazda, and Emil Wiedemann, Weak-strong uniqueness for measure-valued solutions of some compressible fluid models, Nonlinearity 28 (2015), no. 11, 3873–3890. MR 3424896, DOI https://doi.org/10.1088/0951-7715/28/11/3873

Piotr Gwiazda, Ondřej Kreml, and Agnieszka Świerczewska-Gwiazda, Dissipative measure-valued solutions for general conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 3, 683–707. MR 4093617, DOI https://doi.org/10.1016/j.anihpc.2019.11.001

S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Doctoral thesis, Kyoto University, 1984.

J. A. Nohel, R. C. Rogers, and A. E. Tzavaras, Weak solutions for a nonlinear system in viscoelasticity, Comm. Partial Differential Equations 13 (1988), no. 1, 97–127. MR 914816, DOI https://doi.org/10.1080/03605308808820540

Corrado Lattanzio and Athanasios E. Tzavaras, Structural properties of stress relaxation and convergence from viscoelasticity to polyconvex elastodynamics, Arch. Ration. Mech. Anal. 180 (2006), no. 3, 449–492. MR 2214963, DOI https://doi.org/10.1007/s00205-005-0404-3

Corrado Lattanzio and Athanasios E. Tzavaras, Relative entropy in diffusive relaxation, SIAM J. Math. Anal. 45 (2013), no. 3, 1563–1584. MR 3056757, DOI https://doi.org/10.1137/120891307

Tai Ping Liu, Quasilinear hyperbolic systems, Comm. Math. Phys. 68 (1979), no. 2, 141–172. MR 543196

Reza Malek-Madani and John A. Nohel, Formation of singularities for a conservation law with memory, SIAM J. Math. Anal. 16 (1985), no. 3, 530–540. MR 783978, DOI https://doi.org/10.1137/0516038

R. C. MacCamy, A model for one-dimensional, nonlinear viscoelasticity, Quart. Appl. Math. 35 (1977/78), no. 1, 21–33. MR 478939, DOI https://doi.org/10.1090/S0033-569X-1977-0478939-6

Alexey Miroshnikov and Konstantina Trivisa, Relative entropy in hyperbolic relaxation for balance laws, Commun. Math. Sci. 12 (2014), no. 6, 1017–1043. MR 3194369, DOI https://doi.org/10.4310/CMS.2014.v12.n6.a2

Michael Renardy, William J. Hrusa, and John A. Nohel, Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. MR 919738

Denis Serre, The structure of dissipative viscous system of conservation laws, Phys. D 239 (2010), no. 15, 1381–1386. MR 2658332, DOI https://doi.org/10.1016/j.physd.2009.03.014

Denis Serre and Alexis F. Vasseur, $L^2$-type contraction for systems of conservation laws, J. Éc. polytech. Math. 1 (2014), 1–28 (English, with English and French summaries). MR 3322780, DOI https://doi.org/10.5802/jep.1

Luc Tartar, The compensated compactness method applied to systems of conservation laws, Systems of nonlinear partial differential equations (Oxford, 1982) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 111, Reidel, Dordrecht, 1983, pp. 263–285. MR 725524

Athanasios E. Tzavaras, Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of conservation laws, Arch. Rational Mech. Anal. 135 (1996), no. 1, 1–60. MR 1414293, DOI https://doi.org/10.1007/BF02198434

Athanasios E. Tzavaras, Relative entropy in hyperbolic relaxation, Commun. Math. Sci. 3 (2005), no. 2, 119–132. MR 2164193