Skip to main content
Log in

Generalized solutions to Cauchy problems

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

This paper continues previous attempts to find a convenient mathematical setting in which linear and nonlinear Cauchy problems have a unique global solution, that reduces to a classical solution when the latter exists.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Caflish, R. E.: A simplified version of the abstract Cauchy-Kowalevski theorem with weak singularities. Bull. of the AMS23(2), 495–500 (1990).

    Google Scholar 

  2. Colombeau, J. F.: Multiplications of Distributions; a tool in mathematics, numerical engineering and theoretical physics. Lect. Notes Math. 1532. Berlin-Heidelberg-New York: Springer. (1992).

    Google Scholar 

  3. Colombeau, J. F., Heibig, A., Oberguggenberger, M.: Generalized solutions to partial differential equations of evolution type. Preprint.

  4. Heibig, A., Moussaoui, M.: Generalized and classical solutions of nonlinear parabolic equations. J. Nonlinear Analysis, T.M.A. (in press).

  5. John, F.: Partial Differential Equations. Berlin-Heidelberg-New York: Springer. (1986).

    Google Scholar 

  6. Joseph, K. T.: A Riemann problem whose viscosity solutions contain δ measures. Preprint.

  7. Keyfitz, B., Kranzer, H. C.: A viscosity approximation to a system of conservation laws with no classical Riemann solution. Nonlinear Hyperbolic Problems. In: Carasso, C. et al. (eds). pp. 185–197. Lect. Notes Math. 1402. Berlin-Heidelberg-New York: Springer. (1988).

    Google Scholar 

  8. Lafon, F., Oberguggenberger, M.: Generalized solutions to symmetric hyperbolic systems with discontinuous coefficients: the multidimensional case. J. Math. Anal. Appl.160, 93–106 (1991).

    Google Scholar 

  9. Lax, P. D.: The zero dispersion limit, a deterministic analogue of turbulence. Comm. Pure Appl. Math.44, 1047–1056 (1991).

    Google Scholar 

  10. Petrovski, I. G.: Lectures on Partial Differential Equations. New York: Interscience. (1954).

    Google Scholar 

  11. Sanders, R.: On convergence of monotone finite difference schemes with variable space differencing. Math. of Computation40, 91–106 (1983).

    Google Scholar 

  12. Tadmor, E.: Local error estimates for discontinuous solutions of nonlinear hyperbolic equations. SIAM J. Numer. Anal.28, 891–906 (1991).

    Google Scholar 

  13. Trèves, F.: Basic Linear Partial Differential Equations. New York: Academic Press. (1975).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

With 1 Figure

Rights and permissions

Reprints and permissions

About this article

Cite this article

Colombeau, J.F., Heibig, A. Generalized solutions to Cauchy problems. Monatshefte für Mathematik 117, 33–49 (1994). https://doi.org/10.1007/BF01299310

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01299310

1991 Mathematics Subject Classification

Navigation