Advertisement

Applied Mathematics & Optimization

, Volume 78, Issue 2, pp 403–456 | Cite as

On the Existence, Uniqueness and Regularity of Solutions of a Viscoelastic Stokes Problem Modelling Salt Rocks

  • R. A. Cipolatti
  • I.-S. Liu
  • L. A. Palermo
  • M. A. Rincon
  • R. M. S. Rosa
Article

Abstract

A Stokes-type problem for a viscoelastic model of salt rocks is considered, and existence, uniqueness and regularity are investigated in the scale of \(L^2\)-based Sobolev spaces. The system is transformed into a generalized Stokes problem, and the proper conditions on the parameters of the model that guarantee that the system is uniformly elliptic are given. Under those conditions, existence, uniqueness and low-order regularity are obtained under classical regularity conditions on the data, while higher-order regularity is proved under less stringent conditions than classical ones. Explicit estimates for the solution in terms of the data are given accordingly.

Keywords

Generalized Stokes problem Elliptic regularization Viscoelastic fluids Salt modelling 

Mathematics Subject Classification

35Q30 76D06 35B40 37A60 

Notes

Acknowledgements

All the authors acknowledge the financial support of CENPES/PETROBRÁS. R.M.S. Rosa was also partly supported by CNPq, Brasília, Brazil.

References

  1. 1.
    Adams, R.A.: Sobolev Spaces, vol. 2003. Academic Press, Amsterdam (2003)zbMATHGoogle Scholar
  2. 2.
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math. 17, 35–92 (1964)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Agmon, S.: Lectures on Elliptic Boundary Value Problems. Van Nostrand Mathematical Studies. Van Nostrand, Princeton (1965)Google Scholar
  4. 4.
    Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  5. 5.
    Chemin, J.-Y., Masmoudi, N.: About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J. Math. Anal. 33(1), 84–112 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ciarlet, P.G.: Mathematical Elasticity: Three-Dimensional Elasticity, vol 1. Studies in Mathematics & Its Applications. Elsevier, Amsterdam (1988)Google Scholar
  7. 7.
    Evans, L.C.: Partial Differential Equations, vol 19, 2nd edn. Graduate Studies in Mathematics. American Mathematical Society, Providence (2010)Google Scholar
  8. 8.
    Friedman, A.: Partial Differential Equations. Holt, Rinehart and Winston, New York (1969)zbMATHGoogle Scholar
  9. 9.
    Ghidaglia, J.-M.: Régularité des solutions de certains problèmes aux limites linéaires liés aux équations d’Euler. Commun. Partial Differ. Equ. 9(13), 1265–1298 (1984)CrossRefGoogle Scholar
  10. 10.
    Giaquinta, M., Modica, G.: Non linear systems of the type of the stationary Navier–Stokes system. J. Reine Angew. Math. 330, 173–214 (1982)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)zbMATHGoogle Scholar
  12. 12.
    Greene, D.H., Knuth, D.E.: Mathematics for the Analysis of Algorithms. Reprint of the third edition (1990). Modern Birkhäuser Classics. Birkhäuser, Boston (2008)Google Scholar
  13. 13.
    Guillopé, C., Saut, J.-C.: Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal. 15(9), 849–869 (1990)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Haupt, P.: Continuum Mechanics and Theory of Materials, 2nd edn. Springer, New York (2002)CrossRefGoogle Scholar
  15. 15.
    Huy, N.D., Stará, J.: On existence and regularity of solutions to a class of generalized stationary Stokes problem. Comment. Math. Univ. Carol. 47(2), 241–264 (2006)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Lax, P.D., Milgram, A.N.: Parabolic equations. Contributions to the theory of partial differential equations. Ann. Math. Stud. 33, 167–190 (1954)zbMATHGoogle Scholar
  17. 17.
    Lin, F.-H., Liu, C., Zhang, P.: On hydrodynamics of viscoelastic fluids. Commun. Pure Appl. Math. 58(11), 1437–1471 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lin, F.-H., Zhang, P.: On the initial-boundary value problem of the incompressible viscoelastic fluid system. Commun. Pure Appl. Math. 61(4), 539–558 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer, New York (1972)zbMATHGoogle Scholar
  20. 20.
    Lions, P.L., Masmoudi, N.: Global solutions for some Oldroyd models of non-Newtonian flows. Chin. Ann. Math. Ser. B 21(2), 131–146 (2000)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Liu, I.-S.: Continuum Mechanics. Springer, Berlin (2002)CrossRefGoogle Scholar
  22. 22.
    Liu, I.-S.: Entropy flux relation for viscoelastic bodies. J. Elasticity 90, 259–270 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Liu, C., Walkington, N.J.: An Eulerian description of fluids containing visco-hyperelastic particles. Arch. Ration. Mech. Anal. 159(3), 229–252 (2001)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Liu, I.-S., Cipolatti, R.A., Rincon, M.A., Palermo, L.A.: Successive linear approximation for large deformation—instability of salt migration. J. Elasticity 114(1), 19–39 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mácha, V.: On a generalized Stokes problem. Cent. Eur. J. Math. 9(4), 874–887 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Nečas, J.: Equations aux derivees partielles. Presses de Université de Montréal, Montreal (1965)Google Scholar
  27. 27.
    Sideris, T.C., Thomases, B.: Global existence for three-dimensional incompressible isotropic elastodynamics via the incompressible limit. Commun. Pure Appl. Math. 58(6), 750–788 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Solonnikov, V.A.: Initial-boundary value problem for generalized Stokes equations. Math. Bohem. 126(2), 505–519 (2001)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Teixeira, M.G., Liu, I.-S., Rincon, M.A., Cipolatti, R.A., Palermo, L.A.C.: The influence of temperature on the formation of salt domes. Int. J. Model. Simul. Pet. Ind. (Online) 8(2), 35–41 (2014)Google Scholar
  30. 30.
    Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and its Applications, 2nd, 3rd edn. North-Holland, Amsterdam. Reedition in 2001 in the AMS Chelsea Series. AMS, Providence (1984)Google Scholar
  31. 31.
    Temam, R., Miranville, A.: Mathematical Modeling in Continuum Mechanics, 2nd edn. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  32. 32.
    Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics, 3rd edn. Edited and with a preface by Stuart S. Antman. Springer, Berlin (2004)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • R. A. Cipolatti
    • 1
  • I.-S. Liu
    • 1
  • L. A. Palermo
    • 2
  • M. A. Rincon
    • 1
  • R. M. S. Rosa
    • 1
  1. 1.Instituto de MatemáticaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  2. 2.CENPES/PetrobrásRio de JaneiroBrazil

Personalised recommendations