Averaging theorems for conservative systems and the weakly compressible Euler equations |
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Authors: | G Métivier S Schochet |
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Institution: | a IRMAR UMR 6625 CNRS, Université de Rennes I, Campus Beaulieu, 35042 Rennes Cedex, France b School of Mathematical Sciences, Tel Aviv University, 69978 Ramat Aviv, Israel |
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Abstract: | A generic averaging theorem is proven for systems of ODEs with two-time scales that cannot be globally transformed into the usual action-angle variable normal form for such systems. This theorem is shown to apply to certain Fourier-space truncations of the non-isentropic slightly compressible Euler equations of fluid mechanics. For the full Euler equations, we derive formally the generic limit equations and analyze some of their properties. In the one-dimensional case, we prove a generic converic convergence result for the full Euler equations, analogous to the result for ODEs. By making use of special properties of the one-dimensional equations, we prove convergence to the solution of a more complicated set of averaged equations when the genericity assumptions fail. |
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