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An Evans function for 2-D shear flows of the Euler equations on the torus Marangell, Robert


This talk will consider the stability of time independent solutions to the incompressible, inviscid Euler equations on the torus whose stream functions have the form $\psi = U(\xi) =U(p_1x + p_2y)$ for fixed integers $p_1$ and $p_2$. By an appropriate change of coordinates and separation of variables, the linearised spectral problem is reduced to the study of a Hill's equation with a complex potential. By using Hill determinants, an Evans function of the original linearised Euler equation can be constructed. For certain, well-known shear flows, the form of the Hill determinant makes such an Evans function numerically straightforward to compute.

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