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Stability for time-dependent differential equations

Published online by Cambridge University Press:  07 November 2008

Heinz-Otto Kreiss  and
Jens Lorenz
Heinz-Otto Kreiss
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095, USA E-mail: [email protected]
Jens Lorenz
Affiliation:
Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131, USA E-mail: [email protected]
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Extract

In this paper we review results on asymptotic stability of stationary states of PDEs. After scaling, our normal form is ut = Pu + ε f(u, ux,…) + F(x, t), where the (vector-valued) function u(x, t) depends on the space variable x and time t. The differential operator P is linear, F(x, t) is a smooth forcing, which decays to zero for t → ∞, and εf(u, …) is a nonlinear perturbation. We will discuss conditions that ensure u → 0 for t → ∞ when |ε| is sufficiently small. If this holds, we call the problem asymptotically stable.

While there are many approaches to show asymptotic stability, we mainly concentrate on the resolvent technique. However, comparisons with the Lyapunov technique will also be given. The emphasis on the resolvent technique is motivated by the recent interest in pseudospectra.

Type
Research Article
Information
Acta Numerica , Volume 7 , January 1998 , pp. 203 - 285
Copyright
Copyright © Cambridge University Press 1998

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References

REFERENCES

Hagstrom, T. and Keller, H. B. (1986),‘Exact boundary conditions at an artificial boundary for partial differential equations in cylinders’, SIAM J. Math. Anal. 17, 322–341.CrossRefGoogle Scholar
Hagstrom, T. and Lorenz, J. (1995), ‘All-time existence of smooth solutions to PDEs of mixed type and the invariant subspace of uniform states, Adv. Appl. Math. 16, 219–257.CrossRefGoogle Scholar
Henry, D. (1981), Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin.CrossRefGoogle Scholar
Kawashima, S. (1987), ‘Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications’, Proceedings of the Royal Society of Edinburgh 106A, 169–194.CrossRefGoogle Scholar
Klainerman, S. and Ponce, G. (1983), ‘Global, small amplitude solutions to nonlinear evolution equations’, Comm. Pure Appl. Math. 36, 133–141.CrossRefGoogle Scholar
Kreiss, G. and Kreiss, H.-O. (1997), Stability of systems of viscous conservation laws, Technical Report 97–54, UCLA CAM Report.Google Scholar
Kreiss, G., Kreiss, H.-O. and Lorenz, J. (1998a), ‘On stability of conservation laws’, SIAM J. Math. Anal. To appear.Google Scholar
Kreiss, G., Kreiss, H.-O. and Lorenz, J. (1998b), ‘On the use of symmetrizers for stability analysis’. In preparation.Google Scholar
Kreiss, G., Kreiss, H.-O. and Petersson, A. (1994a), ‘On the convergence to steady state of solutions of nonlinear hyperbolic-parabolic systems’, SIAM J. Numer. Anal. 31, 1577–1604.CrossRefGoogle Scholar
Kreiss, G., Lundbladh, A. and Henningson, D. (1994b), ‘Bounds for threshold amplitudes in subcritical shear flow’, J. Fluid Mech. 270, 175–198.CrossRefGoogle Scholar
Kreiss, H.-O. and Lorenz, J. (1989), Initial-Boundary Value Problems and the Navier–Stokes Equations, Academic, New York.Google Scholar
Kreiss, H.-O. and Lui, S. H. (1997), Nonlinear stability of time dependent differential equations, Technical Report 97–43, UCLA CAM Report.Google Scholar
Kreiss, H.-O., Ortiz, O. and Reula, O. (1998c), ‘Stability of quasi-linear hyperbolic dissipative systems’, J. Diff. Eqns. To appear.CrossRefGoogle Scholar
Lyapunov, A. M. (1956), General Problem of the Stability of Motion, Collected Works, Vol. 2, Izdat. Akad. Nauk SSSR, Moscow. (Russian).Google Scholar
Matsumura, A. and Nishida, T. (1979), ‘The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids’, Proc. Jap. Acad., Ser. A 55, 337–341.Google Scholar
Racke, R. (1992), Lectures on Nonlinear Evolution Equations: Initial Value Problems, Vieweg Verlag, Braunschweig, Wiesbaden.CrossRefGoogle Scholar
Romanov, V. A. (1973), ‘Stability and plane-parallel Couette flow’, Funktsional' nyi Analizi Ego Prilozhaniya 7, 62–73. Translated in Funct. Anal. Appl. 7, 137–146.Google Scholar
Sattinger, D. H. (1976), ‘On the stability of waves of nonlinear parabolic systems’, Adv. Math. 22, 312–355.CrossRefGoogle Scholar
Shatah, J. (1982), ‘Global existence of small solutions to nonlinear evolution equations’, J. Diff. Eqns 46, 409–425.CrossRefGoogle Scholar
Strauss, W. (1981), ‘Nonlinear scattering theory at low energy’, J. Fund. Anal. 41, 110–133.CrossRefGoogle Scholar
Trefethen, L. N. (1997), ‘Pseudospectra of linear operators’, SIAM Rev. 39, 383–406.CrossRefGoogle Scholar