Stability of an aneurysm in a membrane tube filled with an ideal fluid

The stability of standing localized structures formed in an axisymmetric membrane tube filled with fluid is studied. It is assumed that the tube wall is heterogeneous and is subjected to localized thinning. Because the problem has no translational invariance in the case of an inhomogeneous wall the stability of a standing localized structure located in the center of the inhomogeneity of the tube wall is understood as the usual, and not the orbital stability up to a shift. The spectral stability of localized elevation waves is considered in the context of aneurysm formation on human vessels. The fluid flowing inside the tube is assumed to be ideal and incompressible and its longitudinal velocity profile is assumed constant along the vertical section of the tube. Spectral stability is established by the proof of the absence of eigenvalues with a positive real part corresponding to exponentially time-increasing perturbations that are solutions of the linearized equations of the problem. The stability analysis is carried out by constructing the Evans function, which depends only on the spectral parameter and is analytic in the right complex half-plane $\Omega^+$. The zeros of the Evans function in $\Omega^+$ coincide with the unstable eigenvalues of the problem. The absence of zeros in $\Omega^+$ is proved by applying the argument principle from complex analysis.