Abstract
In this paper, bifurcation of solutions of a special nonlinear operator equation used in mathematical physics is considered. In the case of an equation for which the Fréchet derivative of the associated operator is a locally perturbed Fredholm operator, sufficient conditions for branching of solutions are studied. The methodology of application of the formalism developed in the paper is demonstrated by the example of the Boltzmann equation.
Similar content being viewed by others
REFERENCES
M. M. Vainberg and V. A. Trenogin, Branching Theory of Solutions of Nonlinear Equations [in Russian], Nauka, Moscow, 1969. English transl.: Wolters-Noordho., Leyden, 1974.
K. Deimling, Nonlinear Functional Analysis, Academic Press, New York, 1985.
B. van der Pol and H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Integral, Cambridge Univ. Press, Cambridge, 1964.
N. Dunford and J. T. Schwartz, Linear Operators. General Theory, Interscience Publ., New York–London, 1958.
N. N. Fimin, Spectral Properties of Linear Operator Pencils [in Russian], Preprint no. 7, M. V. Keldysh Institute of Applied Mathematics (Russian Academy of Sciences), Moscow, 1997.
V. V. Ditkin, “Certain spectral properties of a pencil of linear operators in a Banach space,” Mat. Zametki [Math. Notes], 22 (1977), no. 6, 847–857.
R. E. Caflisch and B. Nicolaenko, “Shock profile solutions of the Boltzmann equation,” Comm. Math. Phys., 86 (1982), 161–194.
E. Zeidler, Nonlinear Functional Analysis and its Applications. Fixed-Point Theorems, Springer-Verlag, New York, 1985.
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin–Heidelberg–New York, 1966.
S. Goldberg, Unbounded Linear Operators, McGraw-Hill, New York, 1966.
V. A. Ditkin, On Spectral Properties of Operator Pencils [in Russian], Computer Center of the Academy of Sciences of the USSR, Moscow, 1983.
A. G. Rutkas, “Cauchy problem for the equation \(Ax'(t) + Bx(t) = f(t)\),” Differentsial nye Uravneniya [Differential Equations], 11 (1975), no. 11, 1996–2010.
V. A. Ditkin, “On certain spectral properties of a pencil of linear bounded operators,” Mat. Zametki [Math. Notes], 31 (1982), no. 1, 75–79.
H. Grad, “Asymptotic theory of the Boltzmann equation. Part 2,” in: Proc. Intern. Symp. Rarefied Gas Dynamics, vol. 1, Academic Press, New York, 1963, pp. 26–59.
B. Nicolaenko, “Dispersion laws for plane wave propagation,” in: The Boltzmann Equation (F. A. Grunbaum, editor), Courant Inst. Math. Sci., New York, 1971.
M. Schechter, “On the essential spectrum of an arbitrary operator,” J. Math. Anal. Appl., 13 (1966), 205–213.
V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow, 1993.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fimin, N.N., Chuyanov, V.A. Branching of Solutions of the Abstract Kinetic Equation. Mathematical Notes 73, 103–109 (2003). https://doi.org/10.1023/A:1022130202626
Issue Date:
DOI: https://doi.org/10.1023/A:1022130202626