Advertisement

Existence Theory of Nonlinear Dissipative Dynamics

  • Viorel BarbuEmail author
Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we present several applications of general theory to nonlinear dynamics governed by partial differential equations of dissipative type illustrating the ideas and general existence theory developed in the previous section. Most of significant dynamics described by partial differential equations can be written in the abstract form (4.1) with appropriate quasi-m-accretive operator A and Banach space X. The boundary value conditions are incorporated in the domain of A. The whole strategy is to find the appropriate operator A and to prove that it is quasi-m-accretive. The main emphasis here is on parabolic-like boundary value problems and the nonlinear hyperbolic equations although the area of problems covered by general theory is much larger.

Keywords

Variational Inequality Maximal Monotone Free Boundary Problem Entropy Solution Stefan Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Alikakos, R. Rostamian, Large time behaviour of solutions of Neumann boundary value problems for the porous medium equations, Indiana Univ. Math. J., 39 (1981), pp. 749–785.CrossRefMathSciNetGoogle Scholar
  2. 2.
    A.N. Antontsev, J.I. Diaz, S. Shmarev, Energy Methods for Free Boundary Problems, Birkhäuser, Basel, 2002.zbMATHGoogle Scholar
  3. 3.
    V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993zbMATHGoogle Scholar
  4. 4.
    V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.zbMATHGoogle Scholar
  5. 5.
    V. Barbu, I. Lasiecka, M. Rammaha, Blow up of generalized solutions to wave equations with nonlinear degenerate damping and source term, Trans. Amer. Math. Soc., 357 (2005), pp. 2571–2611.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    V. Barbu, S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equations, J. Math. Anal. Appl. 255 (2001), 281–307.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    V. Barbu, S. Sritharan, m-accretive quantization of vorticity equation, Semigroup of operators: Theory and applications, Progress in Nonliner Differentiable Equations, 42, 2000, pp. 296–303, Birkhäuser, Basel.Google Scholar
  8. 8.
    Ph. Bénilan, Opérateurs accétifs et semigroupes dans les espaces L p, 1 ≤ p ≤ ∞, Functional Analysis and Numerical Analysis, pp. 15–51, T. Fuzita (Ed.), Japan Soc., Tokyo, 1978.Google Scholar
  9. 9.
    Ph. Bénilan, M.G. Crandall, The continuous dependence on ϕ of solutions of u tΔϕ(u) = 0, Indiana Univ. Math. J., 30 (1981), pp. 161–177.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ph. Benilan, M.G. Crandall, M. Pierre, Solutions of the porous medium equations in R N under optimal conditions on initial values, Indiana Univ. Math. J., 33 (1984), pp. 51–87.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ph. Benilan, N. Kruzkhov, Conservation laws with continuous flux conditions, Nonlinear Differential Eqs. Appl., 3 (1996), pp. 395–419.zbMATHCrossRefGoogle Scholar
  12. 12.
    E. Bonetti, P. Colli, M. Fabrizio, G. Gilardi, Existence and boundedness of solutions for a singular phase field system, J. Differential Eqs., 246 (2009), pp. 3260–3295.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    H. Brezis, Intégrales convexes dans les espaces de Sobolev, Israel J. Math., 13 (1972), pp. 9–23.CrossRefMathSciNetGoogle Scholar
  14. 14.
    H. Brezis, Problèmes unilatéraux, J. Math. Pures Appl., 51 (1972), pp. 1–168.MathSciNetGoogle Scholar
  15. 15.
    H. Brezis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contributions to Nonlinear Functional Analysis, E. Zarantonello (Ed.), Academic Press, New York, 1971.Google Scholar
  16. 16.
    H. Brezis, M.G. Crandall, Uniqueness of solutions of the initial-value problem for u tΔϕ(u) = 0, J. Math. Pures Appl., 58 (1979), pp. 153–163.zbMATHMathSciNetGoogle Scholar
  17. 17.
    H. Brezis, A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl., 62 (1983), pp. 73–97.zbMATHMathSciNetGoogle Scholar
  18. 18.
    G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Ration. Mech. Anal., 92 (1996), pp. 206–245.MathSciNetGoogle Scholar
  19. 19.
    P. Constantin, C. Foias, Navier-Stokes Equations, The University of Chicago Press, Chicago, 1988.zbMATHGoogle Scholar
  20. 20.
    M.G. Crandall, The semigroup approach to the first order quasilinear equations in several space variables, Israel J. Math., 12 (1972), pp. 108–132.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    G. Duvaut, Résolution d'un problème de Stefan, C.R. Acad. Sci. Paris, 267 (1973), pp. 1461–1463.MathSciNetGoogle Scholar
  22. 22.
    G. Duvaut, J.L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.zbMATHGoogle Scholar
  23. 23.
    C.M. Elliott, J.R. Ockendon, Weak and Variational Methods for Moving Boundary Value Problems, Pitman, London, 1992.Google Scholar
  24. 24.
    L.C. Evans, Applications of nonlinear semigroup theory to certain partial differential equations, Proc. Symp. Nonlinear Evolution Equations, M.G. Crandall (Ed.), Academic Press, New York (1978), pp. 163–188.Google Scholar
  25. 25.
    L.C. Evans, Differentiability of a nonlinear semigroup in L 1, J. Math. Anal. Appl., 60 (1977), pp. 703–715.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    M. Fremond, Non-Smooth Thermo-Mechanics, Springer-Verlag, Berlin, 2002.Google Scholar
  27. 27.
    A. Friedman, Variational Principles and Free-Boundary Problems, John Wiley, New York, 1983.Google Scholar
  28. 28.
    A. Haraux, Semilinear Hyperbolic Problems in Bounded Domains, Mathematical Reports, vol. 3, Paris, 1989.Google Scholar
  29. 29.
    Y. Konishi, On the nonlinear semigroups associated with u t = Δβ(u) and π(u t) = Δu, J. Math. Soc., 25 (1973), pp. 622–627.zbMATHMathSciNetGoogle Scholar
  30. 30.
    S.N. Kružkov, First-order quasilinear equations in several independent variables, Math. Sbornik, 10 (1970), pp. 217–236.CrossRefGoogle Scholar
  31. 31.
    O.A. Ladyzenskaya, V.A. Solonnikov, N.N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society Transl., American Mathematical Society, Providence, RI, 1968.Google Scholar
  32. 32.
    I. Lefter, Navier-Stokes equations with potentials, Abstract Appl. Anal., 30 (2007), p. 30.MathSciNetGoogle Scholar
  33. 33.
    J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod, Gauthier–Villars, Paris 1969.Google Scholar
  34. 34.
    G. Marinoschi, A free boundary problem describing the saturated-unsaturated flow in porous medium, Abstract App. Anal., 2005, pp. 813–854.Google Scholar
  35. 35.
    G. Marinoschi, Functional Approach to Nonlinear Models of Water Flows in Soils, Springer, New York, 2006.Google Scholar
  36. 36.
    A. Pazy, The Lyapunov method for semigroups of nonlinear contractions in Banach spaces, Journal d'Analyse Mathématiques, 40 (1982), pp. 239–262.CrossRefMathSciNetGoogle Scholar
  37. 37.
    M. Reed, B. Simon, Methods of Modern Mathematical Physics, American Mathematical Society, New York, 1979.zbMATHGoogle Scholar
  38. 38.
    J. Serrin, G. Todorova, E. Vitillaro, Existence for a nonlinear wave equation with damping and source terms, Diff. Integral Eqs., 16 (2003), pp. 13–50.zbMATHMathSciNetGoogle Scholar
  39. 39.
    R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, SIAM Philadelphia, 1983.Google Scholar
  40. 40.
    J.L. Vasquez, The Porous Medium Equation, Oxford University Press, Oxford, UK, 2006.CrossRefGoogle Scholar
  41. 41.
    V. Veron, Effects regularisants de semigroupes nonlinéaire dans les espaces de Banach, Annales Faculté Sciences Toulouse, 1 (1979), pp. 171–200.zbMATHMathSciNetGoogle Scholar
  42. 42.
    A. Visintin, Differential Models of Hysteresis, Springer-Verlag, Berlin, 1994.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Fac. MathematicsAl. I. Cuza UniversityIasiRomania

Personalised recommendations