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Abstract: Extending recent results in the isentropic case, we use a combination ofasymptotic ODE estimates and numerical Evans-function computations to examinethe spectral stability of shock-wave solutions of the compressibleNavier-Stokes equations with ideal gas equation of state. Our main results arethat, in appropriately rescaled coordinates, the Evans function associated withthe linearized operator about the wave i converges in the large-amplitudelimit to the Evans function for a limiting shock profile of the same equations,for which internal energy vanishes at one endstate; and ii has no unstablepositive real part zeros outside a uniform ball $|\lambda|\le \Lambda$. Thus,the rescaled eigenvalue ODE for the set of all shock waves, augmented with thenonphysical limiting case, form a compact family of boundary-value problemsthat can be conveniently investigated numerically. An extensive numericalEvans-function study yields one-dimensional spectral stability, independent ofamplitude, for gas constant $\gamma$ in $1.2, 3$ and ratio $ u-\mu$ of heatconduction to viscosity coefficient within $0.2,5$ $\gamma\approx 1.4$,$ u-\mu\approx 1.47$ for air. Other values may be treated similarly but werenot considered. The method of analysis extends also to the multi-dimensionalcase, a direction that we shall pursue in a future work.



Autor: Jeffrey Humpherys, Gregory Lyng, Kevin Zumbrun

Fuente: https://arxiv.org/







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