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.
Similar content being viewed by others
References
Caflish, R. E.: A simplified version of the abstract Cauchy-Kowalevski theorem with weak singularities. Bull. of the AMS23(2), 495–500 (1990).
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).
Colombeau, J. F., Heibig, A., Oberguggenberger, M.: Generalized solutions to partial differential equations of evolution type. Preprint.
Heibig, A., Moussaoui, M.: Generalized and classical solutions of nonlinear parabolic equations. J. Nonlinear Analysis, T.M.A. (in press).
John, F.: Partial Differential Equations. Berlin-Heidelberg-New York: Springer. (1986).
Joseph, K. T.: A Riemann problem whose viscosity solutions contain δ measures. Preprint.
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).
Lafon, F., Oberguggenberger, M.: Generalized solutions to symmetric hyperbolic systems with discontinuous coefficients: the multidimensional case. J. Math. Anal. Appl.160, 93–106 (1991).
Lax, P. D.: The zero dispersion limit, a deterministic analogue of turbulence. Comm. Pure Appl. Math.44, 1047–1056 (1991).
Petrovski, I. G.: Lectures on Partial Differential Equations. New York: Interscience. (1954).
Sanders, R.: On convergence of monotone finite difference schemes with variable space differencing. Math. of Computation40, 91–106 (1983).
Tadmor, E.: Local error estimates for discontinuous solutions of nonlinear hyperbolic equations. SIAM J. Numer. Anal.28, 891–906 (1991).
Trèves, F.: Basic Linear Partial Differential Equations. New York: Academic Press. (1975).
Author information
Authors and Affiliations
Additional information
With 1 Figure
Rights 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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01299310