Stability of Gasless Combustion Fronts in One-Dimensional Solids
- 140 Downloads
For gasless combustion in a one-dimensional solid, we show a type of nonlinear stability of the physical combustion front: if a perturbation of the front is small in both a spatially uniform norm and an exponentially weighted norm, then the perturbation stays small in the spatially uniform norm and decays in the exponentially weighted norm, provided the linearized operator has no eigenvalues in the right half-plane other than zero. Using the Evans function, we show that the zero eigenvalue must be simple. Factors that complicate the analysis are: (1) the linearized operator is not sectorial, and (2) the linearized operator has good spectral properties only when the weighted norm is used, but then the nonlinear term is not Lipschitz. The result is nevertheless physically natural. To prove it, we first show that when the weighted norm is used, the semigroup generated by the linearized operator decays on a subspace complementary to the operator’s kernel, by showing that it is a compact perturbation of the semigroup generated by a more easily analyzed triangular operator. We then use this result to help establish that solutions stay small in the spatially uniform norm, which in turn helps establish nonlinear convergence in the weighted norm.
Unable to display preview. Download preview PDF.
- Bates, P.W., Jones, C.K.R.T.: Invariant manifolds for semilinear partial differential equations. Dynamics reported 2, 1–38, Dynam. Report. Ser. Dynam. Systems Appl., Vol. 2. Wiley, Chichester, 1989Google Scholar
- Berestycki, H., Larrouturou, B., Roquejoffre, J.-M.: Mathematical investigation of the cold boundary difficulty in flame propagation theory. Dynamical Issues in Combustion Theory, Vol. 35 (Minneapolis, MN, 1989) IMA Vol. Math. Appl., Springer, New York, 37–61, 1991Google Scholar
- Chicone, C., Latushkin, Y.: Evolution Semigroups in Dynamical Systems and Differential Equations. Math. Surv. Monogr. Vol. 70. AMS, Providence, 1999Google Scholar
- Evans, J.W.: Nerve axon equations. III. Stability of the nerve impulse. Indiana Univ. Math. J. 22, 577–593(1972/73)Google Scholar
- Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, Vol. 840. Springer, Berlin-New York, 1981Google Scholar
- Sandstede, B.: Stability of travelling waves. Handbook of Dynamical Systems, Vol. 2. North-Holland, Amsterdam, 983–1055, 2002Google Scholar
- Sandstede, B., Scheel, A. Absolute and convective instabilities of waves on unbounded and large bounded domains. Physica D 145, 233–277Google Scholar
- Sell, G., You, Y.: Dynamics of evolutionary equations. Applied Mathematical Sciences, Vol. 143, Springer, New York, 2002Google Scholar