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We study a complex Ginzburg-Landau (GL) type model related to fluidinstabilities in the boundary of magnetized toroidal plasmas (callededge-localized modes) with a prescribed shear flow on the Neumann boundarycondition. We obtain the following universal results for the model in aone-dimensional interval. First, if the shear is weak, there is a uniquelinearly stable steady-state perturbed from the nonzero constant steady-statecorresponding to the zero shear case. Second, if the shear is strong, there isno plausible steady-state except the trivial zero solution in the interval.With the help of these results and the existence of global attractors, we candramatically reduce the number of cases for the long-time behavior of asolution in the model.

Mathematical analysis of long-time behavior of magnetized fluid instabilities with shear flow