Asymptotic estimates and stability analysis of Kuramoto-Sivashinsky type models |
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Authors: | Milena Stanislavova Atanas Stefanov |
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Institution: | 1. Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS, 66045-7523, USA
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Abstract: | We first show asymptotic L
2 bounds for a class of equations, which includes the Burger-Sivashinsly model for odd solutions with periodic boundary conditions.
We consider the conditional stability of stationary solutions of Kuramoto-Sivashinsky equation in the periodic setting. We
establish rigorously the general structure of the spectrum of the linearized operator, in particular the linear instability
of steady states. In addition, we show conditional asymptotic stability with asymptotic phase, under a natural spectral hypothesis
for the corresponding linearized operator. For the zero solution, we have more precise results. Namely, in the non-resonant
regime L ≠
n
π, we prove conditional asymptotic stability, provided one considers only mean value zero data. If, however, L = n
0
π (but still
ò\nolimits-LL u0(x) dx=0{\int\nolimits_{-L}^L u_0(x) dx=0}), then we have conditional orbital stability. More specifically, the solutions relax to a small (but generally non-zero)
function as long as the initial data are small and lie on a center-stable manifold of codimension 2(n
0 − 1). |
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Keywords: | |
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