ESTIMATION OF THE LINEAR TRANSIENT GROWTH OF PERTURBATIONS OF CELLULAR FLAMES

In this work the rates of the linear transient growth of the perturbations of cellular flames governed by the Sivashinsky equation are estimated. The possibility and significance of such a growth was indicated earlier in both computational and analytical investigations. A numerical investigation of the 2-norm of the resolvent of the linear operator associated with the Sivashinsky equation linearized in a neighborhood of the steady coalescent pole solution was undertaken. The results are presented in the form of the pseudospectra and the lower bound of possible transient amplification. This amplification is strong enough to make the round-off errors visible in the numerical simulations in the form of small cusps appearing on the flame surface randomly in time. The performance of available numerical approaches was compared to each other and the results are checked versus directly calculated norms of the evolution operator.

Boundedness of solutions of matrix nonlinear volterra difference equations

Nonlinear discrete-time Volterra equations in a Euclidean space are considered. Conditions for the boundedness of solutions are established by virtue of recent estimates for the norm of the resolvent of Volterra operators. The conditions are formulated in the terms of coefficients of considered equations. In addition, estimates for thec0- andl2-norms of solutions are derived.

Bounds for threshold amplitudes in subcritical shear flows

A general theory which can be used to derive bounds on solutions to the Navier-Stokes equations is presented. The behaviour of the resolvent of the linear operator in the unstable half-plane is used to bound the energy growth of the full nonlinear problem. Plane Couette flow is used as an example. The norm of the resolvent in plane Couette flow in the unstable half-plane is proportional to the square of the Reynolds number (R). This is now used to predict the asymptotic behaviour of the threshold amplitude below which all disturbances eventually decay. A lower bound is found to be R−21/4. Examples, obained through direct numerical simulation, give an upper bound on the threshold curve, and predict a threshold of R−1. The discrepancy is discussed in the light of a model problem.

Order of singularity applied to linear operators

SynopsisWe extend the notion of order of singularity for complex-valued functions, as originally set forth by Hadamard, to Banach algebra-valued functions. Restricting our attention to linear operators whose spectral radius is one, we obtain a connection between the rate of growth of the norm of iterates of a linear operator and the rate of growth of the norm of the resolvent of the operator near the spectrum of the operator. In the finite dimensional case, we obtain an upper bound on the size of the Jordan block corresponding to an eigenvalue of maximum modulus.

31.—Spectral Manifolds for Constant Coefficient Elliptic Operators in Lp(Rn)

SynopsisThe principal results of this paper concern the spectral properties of the maximal realisation Pp in Lp(Rn) of a formally self adjoint constant coefficient strongly elliptic partial differential operator P(D), assumed to be homogeneous of order 2m, for 1 ≦ p ≦ ∞ and n ≧ 2. If we assume that , for 2n/n + l ≦ p ≦ 2n/n−1, together with certain assumptions on the associated real zero surfaces P(ξ) = λ, λ > 0, then σ(Pp) = σc(Pp) = [0, ∞). We obtain an estimate on the norm of the resolvent of Pp for points near the real axis, which allows us to establish the existence of a generalised resolution of the identity in the sense of Kocan.