Abstract
This report outlines the role of the weak topology and the representation of weak limits by Young measures in proving the trend of dissipative infinite dimensional dynamical systems to equilibria. Two examples are provided: (i) a weakly damped wave equation and (ii) an infinite system of ordinary differential equations which model a liquid vapor phase transition.
This research was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Contract/Grant No. AFOSR-87-0315. The United States Government is authorized to reproduce and distribute reprints for government purposes not withstanding any copyright herein.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
J. Hale, “Asymptotic Behavior of Dissipative Systems”, American Mathematical Society, Providence, RI, 1988.
R. Temam, “Infinite-Dimensional Dynamical Systems in Mechanics and Physics”, Applied Mathematical Sciences 68, Springer Verlag, New York Berlin Heidelberg, 1988.
M. Slemrod, Weak asymptotic decay via a “relaxed invariance principle” for a wave equation with nonlinear, nonmonotone damping, to appear Proc. Royal Society of Edinburgh.
M. Slemrod, Trend to equilibrium in the Becker-Döring cluster equations, to appear Nonlinearity.
C. M. Dafermos, Asymptotic behavior of some evolutionary systems, in “Nonlinear Evolution Equations” (M. G. Crandall, ed.) pp. 141–154, Academic Press, New York, 1978.
A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differential Equations 59 (1985), 145–154.
M. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. in Partial Differential Equations 7 (1982), 959–1000.
J. M. Ball, A version of the fundamental theorem for Young measures, to appear in Proc. of CNRS-NSF Conference on Continuum Models of Phase Transitions, Springer Lecture Notes in Mathematics (ed. D. Serre, M. Rascle, M. Slemrod), 1990.
L. Tartar, Compensated compactness and applications to partial differential equations, in “Research Notes in Mathematics 39; Nonlinear Analysis and Mechanics: Heriot Watt Symposium, Vol. IV” (R. J. Knops, ed.) pp. 136–211, Pitman Press, London, 1975.
L. C. Young, “Lectures on the Calculus of Variations and Optimal Control Theory”, W. B. Saunders Co., Philadelphia, London, 1969.
O. Penrose and J. L. Lebowitz, Towards a rigorous molecular theory of metastability, in “Studies in Statistical Mechanics VII, Fluctuation Phenomena” (E. Montroll and J. L. Lebowitz, ed.), North Holland, 1976.
R. Becker and W. Wring, Kinetische Behandlung der Keimbildung in übersattigten Dämpfer, Ann. Phys. (Leipzig) 24 (1935), 719–752.
J. M. Ball, J. Carr, O. Penrose, The Becker-Wring cluster equations: basic properties and asymptotic behavior of solutions, Comm. Math. Physics 104 (1986), 657–692.
J. M. Ball and J. Carr, Asymptotic behavior of solutions to the Becker-Wring equations for arbitrary initial data, Proc. Royal Society of Edinburgh 108A (1988), 109–116. *** DIRECT SUPPORT *** A3418285 00003
Author information
Authors and Affiliations
Editor information
Additional information
Dedicated to Jack Hale on the occasion of his 60th birthday.
Rights and permissions
Copyright information
© 1990 Springer-Verlag
About this paper
Cite this paper
Slemrod, M. (1990). The relaxed invariance principle and weakly dissipative infinite dimensional dynamical systems. In: Kirchgässner, K. (eds) Problems Involving Change of Type. Lecture Notes in Physics, vol 359. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-52595-5_87
Download citation
DOI: https://doi.org/10.1007/3-540-52595-5_87
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-52595-0
Online ISBN: 978-3-540-47049-6
eBook Packages: Springer Book Archive