Israel Journal of Mathematics

, Volume 12, Issue 2, pp 108–132 | Cite as

The semigroup approach to first order quasilinear equations in several space variables

  • Michael G. Crandall


The Cauchy problem for u t + Σ i = 1 n (φ i (u)) xi = 0 is treated via the theory of semigroups of nonlinear transformations. This treatment requires the development of results concerning the time-independent equation u + Σ i = 1 n (φ i (u)) xi = h for hL 1(Rn), which in turn is studied via the regularized equation
$$ u + \sum\nolimits_{i = 1}^n {\left( {\phi _i \left( u \right)} \right)} _{xi} - \varepsilon \Delta u = h $$


Generalize Solution Cauchy Problem Generation Theorem Nonlinear Transformation QUASILINEAR Equation 


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  1. 1.
    Ph. Benilan,Solutions intégrales d’equations d’évolution dans un espace de Banach, C.R. Acad. Sci. Paris,274 (1972), 47–50.MATHMathSciNetGoogle Scholar
  2. 2.
    H. Brezis,Perturbations non linéaires d’operateurs maximaux monotones, C. R. Acad. Sci. Paris269 (1969), 566–569.MATHMathSciNetGoogle Scholar
  3. 3.
    H. Brezis andW. Strauss, to appear.Google Scholar
  4. 4.
    E. Conway andJ. Smoller,Global solutions of the Cauchy problem for quasilinear first order equations in several space variables, Comm. Pure Appl. Math.19 (1966), 95–105.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    M. G. Crandall,A generalized domain for semigroup generators, Proc. Amer. Math. Soc., to appear.Google Scholar
  6. 6.
    M. G. Crandall,Semigroups of nonlinear transformations in Banach spaces, Contributions to Nonlinear Functional Analysis, 157–179, Educardo H. Zarantonello, Editor, Academic Press, New York and London, 1971.Google Scholar
  7. 7.
    M. G. Crandall, MRC Technical Summary Report, in preparation.Google Scholar
  8. 8.
    M. G. Crandall andT. M. Liggett,Generation of semi-groups of nonlinear transformation on general Banach spaces, Amer. J. Math.,93 (1971), 265–298.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    M. G. Crandall andA. Pazy,Nonlinear evolution equations in Banach spaces, Israel J. Math.11 (1972), 57–94.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    A. Douglis,Layering methods for nonlinear partial differential equations of first order, Ann. Inst. Fourier.Google Scholar
  11. 11.
    T. Kato,Nonlinear semi-groups and evolution equations, J. Math. Soc. Japan19 (1967), 508–520.MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Y. Konishi,Nonlinear semigroups in Banach lattices, Proc. Japan Acad.47 (1971), 24–28.MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    S. N. Kružkov,First order quasilinear equations in several independent variables. Math. USSR-Sb.10 (1970), 217–243.CrossRefGoogle Scholar
  14. 14.
    B. Quinn,Solutions with shocks: An example of an L1-contractive semigroup Comm. Pure Appl. Math.24 (1971), 125–132.CrossRefMathSciNetGoogle Scholar
  15. 15.
    G. Stampacchia,Equations Elliptiques du Second Ordre a Coefficients Discontinus, Les Presses de L’Université de Montreal, Montreal, 1966.MATHGoogle Scholar
  16. 16.
    A. I. Vol’pert,The spaces BV and quasilinear equations, Math. USSR-Sb.,2 (1967), 225–267.MATHCrossRefGoogle Scholar

Copyright information

© Hebrew University 1972

Authors and Affiliations

  • Michael G. Crandall
    • 1
  1. 1.Mathematics Research CenterUniversity of WisconsinUSA

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