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On certain mathematical models in continuum thermomechanics

  • V. G. Zvyagin
  • V. P. Orlov
Article

Abstract

This paper presents results on solvability of multidimensional systems of equations of thermoviscoelasticity. Both compressible and incompressible continua are considered. The existence and uniqueness of regular, weak, and weak-renormalized solutions are given, both local and global. The presented results are based on the successive approximation method combined with an a priori estimates, a fixed point argument and passage to the limit technique. The theory of anisotropic Sobolev spaces with a mixed norm and abstract differential equations are used.

Mathematics Subject Classification

Primary 80A17 Secondary 35Q35 

Keywords

Thermoviscoelastic continuum a priori estimates successive approximations fixed point theorem weak solution weak-renormalized solution Oberbeck–Boussinesq-type system objective Jaumann derivative 

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References

  1. 1.
    Agranovich Y.Y., Sobolevskii P.E.: Motion of nonlinear visco-elastic fluid. Nonlinear Anal. 32, 755–760 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Allen S.M., Cahn J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)CrossRefGoogle Scholar
  3. 3.
    Alt H.M., Luckhaus S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183, 311–341 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory. Birkhüser, Boston, MA, 1995.Google Scholar
  5. 5.
    Anderson D.M., McFadden G.B., Wheeler A.A.: A phase-field model of solidification with convection. Phys. D 135, 175–194 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    S.N.Antontsev, A.V.Kazhikhov and V.N.Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. Studies in Mathematics and Its Applications 22, North–Holland, Amsterdam, 1990.Google Scholar
  7. 7.
    G. Astarita and G. Marucci, Principles of Non-Newtonian Fluid Hydromechanics. McGraw–Hill, New York, 1974.Google Scholar
  8. 8.
    A. Attaoui, D. Blanchard and O. Guibé, Weak-renormalized solution for a nonlinear Boussinesq system. Differential Integral Equations 22 (2009), 465–494.Google Scholar
  9. 9.
    Beckermann C.: Modeling melt convection in phase-field simulations of solidification. J. Comput.Phys. 154, 468–496 (1999)CrossRefzbMATHGoogle Scholar
  10. 10.
    D. Blanchard, A few result on coupled systems of thermomechanics. In: On the Notions of Solution to Nonlinear Elliptic Problems: Results and Developments, Quad. Mat. 23, 2008, 145–182.Google Scholar
  11. 11.
    D. Blanchard, N. Bruyere and O. Guibé, Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Commun. Pure Appl. Anal. 12 (2013), 2213–2227.Google Scholar
  12. 12.
    D. Blanchard and O. Guibé, Existence of solution for a nonlinear system in thermoviscoelasticity. Adv. Differential Equations 5 (2000), 1221–1252.Google Scholar
  13. 13.
    Blanchard D., Murat F.: Renormalized solution for nonlinear parabolic problems with L 1 data, existence and uniqueness. Proc. Roy. Soc. Edinburgh Sect. A 127, 1137–1152 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    D. Blanchard, F. Murat and H. Redwane, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems. J. Differential Equations 177 (2001), 331–374.Google Scholar
  15. 15.
    D. Blanchard, F. Murat and H. Redwane, Renormalized solutions for a class of nonlinear evolution problems. J. Math. Pures Appl. (9) 77 (1998), 117–151.Google Scholar
  16. 16.
    L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87 (1989), 149–169.Google Scholar
  17. 17.
    L. Boccardo, A. Dall’Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data. J. Funct. Anal. 147 (1997), 237–258.Google Scholar
  18. 18.
    Bonetti E., Bonfanti G.: Existence and uniqueness of the solution to a 3D thermoelastic system. Electron. J. Differential Equations 50, 1–15 (2003)MathSciNetGoogle Scholar
  19. 19.
    Bonetti E., Colli P., Laurençot P.: Global existence for a hydrogen storage model with full energy balance. Nonlinear Anal. 75, 3558–3573 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    M. Bulíček, R. Lewandowski and J. Málek, On evolutionary Navier-Stokes-Fourier type systems in three spatial dimensions. Comment. Math. Univ. Carolin. 52 (2011), 89–114.Google Scholar
  21. 21.
    M. Bulíček, J. Málek and K. R. Rajagopal, Mathematical analysis of unsteady flows of fluids ith pressure, shear-rate, and temperature dependent material moduli that slip at solid boundaries. SIAM J. Math. Anal. 41 (2009), 665–707.Google Scholar
  22. 22.
    G. Caginalp, An analysis of phase field model of a free boundary. Arch. Ration. Mech. Anal. 92 (1986), 205–245.Google Scholar
  23. 23.
    G. Caginalp and J. Jones, A derivation and analysis of phase field models of thermal alloy. Ann. Phys. 237 (1995), 66–107.Google Scholar
  24. 24.
    J. Cahn and J. Hilliard, Free energy of a nonuniform system: I. Interfacial free energy. J. Chem. Phys. 28 (1958), 258–267.Google Scholar
  25. 25.
    Consiglieri L.: Weak solution for a class of non-Newtonian fluids with energy transfer. J. Math. Fluid Mech. 2, 267–293 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Consiglieri L.: Friction boundary conditions on thermal incompressible viscous flows. Ann. Mat. Pura Apll. 187(4), 647–665 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Consiglieri L.: Regularity for the Navier-Stokes-Fourier system. Differ. Equ. Appl. 1, 583–604 (2009)zbMATHMathSciNetGoogle Scholar
  28. 28.
    B. Climent and E. Fernández-Cara, Existence and uniqueness results for a coupled problem related to the stationary Navier-Stokes system. J. Math. Pures Appl. (9) 76 (1997), 307–319.Google Scholar
  29. 29.
    B. Climent and E. Fernández-Cara, Some existence and uniqueness results for a time-dependent coupled problem of the Navier-tokes kind. Math. Models Methods Appl. Sci. 8 (1998), 603–622.Google Scholar
  30. 30.
    Dall’Aglio A., Orsina L.: Nonlinear parabolic equations with natural growth conditions and L 1 data. Nonlinear Anal. 27, 59–73 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Denk R., Hieber M., Prüss J.: Optimal L pL q-estimates for parabolic boundary value problems with inhomogeneous data. Math. Z. 257, 193–224 (2007)zbMATHGoogle Scholar
  32. 32.
    Díaz J.I., Galiano G.: Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion. Topol. Methods Nonlinear Anal. 11, 59–82 (1998)zbMATHMathSciNetGoogle Scholar
  33. 33.
    J.I. Díaz, J.-M. Rakotoson and P. G. Schmidt, A parabolic system involving a quadratic gradient term related to the Boussinesq approximation. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 101 (2007), 113–118.Google Scholar
  34. 34.
    R.-J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. of Math. (2) 130 (1989), 321– 366.Google Scholar
  35. 35.
    R.-J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511–547.Google Scholar
  36. 36.
    C. Eck, J. Jarušek and M. Krbec, Unilateral Contact Problems: Variational Methods and Existence Theorems. Pure and Applied Mathematics 270, Chapman and Hall/CRC, Boca Raton, FL, 2005.Google Scholar
  37. 37.
    E. Feireisl and J. Málek, On the Navier-tokes equations with temperature-dependent transport coefficients. Differ. Equ. Nonlinear Mech. 2006 (2006), 1– 14, Article ID 90616.Google Scholar
  38. 38.
    Fernández-Cara E., Vaz C.: Weak-renormalized solutions for a system that models non-isothermal solidification. SeMAJ. 59, 5–18 (2012)CrossRefzbMATHGoogle Scholar
  39. 39.
    M. Frémond, Non-Smooth Thermomechanics. Springer-Verlag, Berlin, 2001.Google Scholar
  40. 40.
    Hoffmann K.-H., Zochowski A.: Existence of solutions to some nonlinear thermoelastic system with viscosity. Math. ech. Appl. Sci. 15, 187–204 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    D. Joseph, Stability of Fluid Motions. I. Springer Tracts Nat. Philos. 27, Springer-Verlag, Berlin, 1976.Google Scholar
  42. 42.
    D. Joseph, Stability of fluid motions. Vol. II. Springer Tracts Nat. Philos. 28, Springer-Verlag, Berlin, 1976.Google Scholar
  43. 43.
    Y. Kagei, M. Ružička and G. Thäter, Natural convection with dissipative heating. Comm. Math.Phys. 214 (2000), 287–313.Google Scholar
  44. 44.
    M. A. Koltunov, A. S. Kravchuck and V. P. Mayboroda, Applied Mechanics of Deformable Solid Body.Vysšaya Skola, Moskow, 1983.Google Scholar
  45. 45.
    M. A. Krasnosel’skiĭ, P. P. Zabreĭko, E. I. Pustyl’nik and P. E. Sobolevskiĭ, Integral Operators in Spaces of Summable Functions. Noordhoff International publishing, Leyden, 1976.Google Scholar
  46. 46.
    S. G. Kreĭn (ed.), Functional Analysis. Nauka, Moscow, 1972.Google Scholar
  47. 47.
    Krylov N.V.: The Calderón-ygmund theorem and its application to parabolic equations. Algebra i Analiz 13, 1–25 (2001)Google Scholar
  48. 48.
    J.-L. Lions, Quelles metodes de resolutinon des problemes aux limites non lineares. Dunod Gauthier-Villar, Paris, 1969.Google Scholar
  49. 49.
    J.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Models. Oxford Lecture Ser. Math. Appl. 3, The Clarendon Press, Oxford University Press, New York, 1996.Google Scholar
  50. 50.
    J. M. Milhaljan, A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid. Astronom. J. 136 (1962), 1126–1133.Google Scholar
  51. 51.
    Moroşanu C., Motreanu D.: A generalized phase-field system. J. Math. Anal. Appl. 273, 515–540 (1999)CrossRefGoogle Scholar
  52. 52.
    Naumann J.: On the existence of weak solutions to the equations of non-stationary motion of heat-conducting incompressible viscous fluids. Math. Methods Appl. Sci. 29, 1883–1906 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  53. 53.
    Nečas J., Roubíček T.: Buoyancy-driven viscous flow with L 1-ata. Nonlinear Anal. 46, 737–755 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    Niezgódka M., Zheng S., Sprekels J.: Global solutions to a model of structural phase transitions in shape memory alloys. J. Math. Anal. Appl. 130, 39–54 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  55. 55.
    Pawłow I., Zaja̧czkowski W.: Global regular solutions to a Kelvin-Voigt type thermoviscoelastic system. SIAM J. Math. Anal. 45, 1997–2045 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  56. 56.
    I. Pawłow and W. Zaja̧czkowski, Unique solvability of a nonlinear thermoviscoelasticity system in Sobolev space with a mixed norm. Discrete Cont. Dyn. Syst. Ser. S 4 (2011), 441–466.Google Scholar
  57. 57.
    Porretta A.: Existence results for nonlinear parabolic equations via strong convergence of truncations. Ann. Mat. Pura Appl. 177(4), 143–172 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  58. 58.
    Racke R., Zheng S.: Global existence and asymptotic behavior in nonlinear thermoviscoelasticity. J. Differential Equations 134, 46–67 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  59. 59.
    K. R. Rajagopal and M. Ružička, Mathematical modeling of electrorheological materials. Continuum Mech. Thermodyn. 13 (2001), 59–78.Google Scholar
  60. 60.
    M. Reiner, Rheology. In: Handbuch der Physik, S. Flügge (ed.), Bd. VI, Springer-Verlag, Berlin, 1958, 434–550.Google Scholar
  61. 61.
    T. Roubíček, Thermo-visco-elasticity at small strains with L 1-data. Quart. Appl. Math. 67 (2009), 47–71.Google Scholar
  62. 62.
    Roubíček T.: On non-Newtonian fluids with energy transfer. J. Math. Fluid Mech. 11, 110–125 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  63. 63.
    Roubíček T.: Nonlinearly coupled thermo-visco-elasticity. NoDEA Nonlinear Differential Equations Appl. 20, 1243–1275 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  64. 64.
    T. Roubíček and G. Tomassetti, Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis. Arch. Ration. Mech. Anal. 210 (2013), 1–43.Google Scholar
  65. 65.
    Sobolevskiĭ P.E.: Coerciveness inequalities for abstract parabolic equations. Dokl. Akad. Nauk SSSR 157, 52–55 (1964)MathSciNetGoogle Scholar
  66. 66.
    Sobolevskiĭ P.E.: Equations of parabolic type in a Banach space. Tr. Mosk. Mat. Obs. 10, 297–351 (1961)Google Scholar
  67. 67.
    Solonnikov V.A.: A priori estimates for solutions of second-order equations of parabolic type. Tr. Mat. Inst. Steklova 70, 133–212 (1964)zbMATHMathSciNetGoogle Scholar
  68. 68.
    Shibata Y.: Global in time existence of small solutions of nonlinear thermoviscoelastic equations. Math. Methods Appl. Sci. 18, 871–895 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  69. 69.
    Talhouk R.: Existence locale et unicité d’écoulements de fluides viscoélastiques dans des domaines non bornés. C. R. Acad. Sci. Paris. Ser. I 328, 87–92 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  70. 70.
    R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis. North– Holland, Amsterdam, 1977.Google Scholar
  71. 71.
    H. Triebel, Interpolation theory, Function Spaces, Differential Operators. North–Holland, Amsterdam, 1978.Google Scholar
  72. 72.
    C. Truesdell, A First Course in Rational Continuum Mechanics. Academic Press, New York, 1977.Google Scholar
  73. 73.
    Turbin M.V., Zvyagin V.G.: Mathematical Question of Hydrodynamcs of Viscoelastic Continua. Krasand, Moscow (2012)Google Scholar
  74. 74.
    Vorotnikov D.A., Zvyagin V.G.: On the solvability of the initial-value problem for the motion equations of nonlinear viscoelastic medium in the whole space. Nonlinear Anal. 58, 631–656 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  75. 75.
    Zvyagin V.G., Vorotnikov D.A.: Approximating-topological methods in some problems of hydrodynamics. J. Fixed Point Theory Appl. 3, 23–49 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Research Institute of MathematicsVoronezh State UniversityVoronezhRussia
  2. 2.Department of MathematicsVoronezh State UniversityVoronezhRussia

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