Summary: The equations \[ x_t=x_s\times x_{ss}+a\Bigl\{x_{sss}+ {\textstyle{3\over2}} x_{ss}\times(x_s\times x_{ss})\Bigr\} \] which describe the motion of a vertex filament with or without an axial flow inside its core are considered. The initial and the initial-boundary value problems are proved to have unique and smooth solutions globally in time. These results are obtained by adding vanishing parabolic terms which conserve the length of the filament.