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It is known that the motion of a vortex filament with axial flow in a perfect fluid is approximately described by a generalization of the localized induction equation. The unique solvability of the initial value problem for it is first established by parabolic regularization. As the axial flow effect vanishes, its solution converges to that of the localized induction equation. Analogous results are obtained in the spatially periodic case.doi:10.2977/prims/1195145146 fatcat:zbtpdlhjxzbkvnela424e4o6om