Solvability of an Initial-Boundary Value Problem for a Second Order Parabolic System with a Third Order Dispersion Term

TL;DR

This paper proves the solvability of an initial-boundary value problem for a second order parabolic system with a third order dispersion term, relevant to vortex filament motion, using a novel regularization technique.

Contribution

It introduces a new regularization method involving a space-time derivative to establish solvability of a complex parabolic-dispersive system.

Findings

Proves solvability of the initial-boundary value problem.

Develops a new regularization technique for dispersive systems.

Applicable to modeling vortex filament motion.

Abstract

We consider a linear second order parabolic system with a third order dispersion term. This type of system arises when considering a nonlinear model equation describing the motion of a vortex filament with axial flow immersed in an incompressible and inviscid fluid. We prove the solvability of an initial-boundary value problem of the parabolic-dispersive system which allows application to the motion of a vortex filament. To do so, we propose a new regularization technique by adding a space-time derivative term.