On the linear stability of hyperbolic PDEs and viscoelastic flows Authors
Received: 16 December 1993 Revised: 11 April 1994 DOI:
Cite this article as: Renardy, M. Z. angew. Math. Phys. (1994) 45: 854. doi:10.1007/BF00952081 Abstract
The issue addressed in this paper is whether linear stability can be determined from the spectrum. We present a counterexample for a hyperbolic PDE in two dimensions and a positive result for parallel shear flows of a class of viscoelastic fluids.
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math., 17, 35–92 (1964).
A. F. Neves, H. de Souza Ribeiro and O. Lopes,
On the spectrum of evolution operators generated by hyperbolic systems, J. Func. Anal., 67, 320–344 (1986).
J. Prüß, On the spectrum of C
o-semigroups, Trans. Amer. Math. Soc., 284, 847–857 (1984).
On the stability of parallel shear flow of an Oldroyd B fluid, Diff. Integral Eq., 6, 481–489 (1993).
M. Renardy, On the type of certain C
o-semigroups, Comm. Part. Diff. Eq., 18, 1299–1307 (1993).
J. Zabczyk, A note on C
o-semigroups, Bull. Acad. Polon. Sci., Ser. Sci. Math., 23, 895–898 (1975).