Linear Stability with Respect to the Initial Value Perturbations in the Presence of Solutions of the Linearized Equation Having Strictly Positive Exponential Growth Rate
Journal Title: INCAS BULLETIN - Year 2016, Vol 8, Issue 1
Abstract
The objective is to emphasize the importance of the functional framework specification when the linear stability of a gas flow is analyzed. For this the linear stability with respect to the initial value perturbations of the constant 1-D gas flow is presented in three different functional frameworks. For the linearized Euler equations in each framework the initial value problem is well-posed in sense of Hadamard and the zero solution is stable, i.e. the constant flow is linearly stable. However, in the first framework the set of the exponential growth rate of the solutions of the linearized equations is the whole real axis and the Briggs-Bers stability analysis can not be applied. In the second framework even if the exponential growth rate of the solutions of the linearized equation is equal to zero, the Briggs-Bers stability analysis fails, because the Fourier transform can not be applied. In the third framework the exponential growth rate of the solutions of the linearized equation is equal to zero, the Briggs-Bers stability analysis works, but there are solutions which satisfy the linearized equations only in a generalized sense (almost everywhere). These considerations can be useful in a better understanding of some apparently strange results obtained in different mathematical models of the sound propagation in a gas flowing in a lined duct.
Authors and Affiliations
Stefan BALINT, Agneta BALINT
Keywords
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- EP ID EP117709
- DOI 10.13111/2066-8201.2016.8.1.4
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