Transport of Rough Vector Fields

While the transport of a scalar quantity advected by a rough vector field (in the Sobolev or BV sense) is fairly well understood, there are relatively few results concerning the transport of rough vector fields themselves. This theory is particularly relevant to the behavior of magnetic field lines "frozen" into an ideal plasma. In the smooth setting, such behavior leads to classical results like Alfvén's Theorem. However, when the fluid motion is governed by a rough vector field, some of these classical results may no longer hold. It is therefore essential to investigate whether well-known properties of magnetic field lines—and concepts such as the Lie bracket or magnetic flux—can be extended to this weaker, less regular context.