  We apply the borderline Sobolev inequalities of Bourgain-Brezis to the
vorticity equation and Navier-Stokes equation in 2D. We take the initial
vorticity to be in the space of functions of Bounded variation(BV). We obtain
the subsequent vorticity to be in the space of functions of bounded variation,
uniformly for small time, and the velocity vector to be uniformly bounded for
small time. Such a conclusion cannot follow for initial vorticity taken to be
just a measure or in L^1 from the Lamb-Oseen vortex example.
  Secondly we apply an improved Strichartz inequality obtained earlier by the
first and third authors to the Maxwell equations of Electromagnetism. In
particular we estimate the size of the magnetic field vector in terms of the
gradient of the current density vector. The main point is that in this
inequality only the L^1 norm in space appears for the gradient of the current
density vector. Such a result is only possible because of a vanishing
divergence inhomogeneity in the wave equation for the Magnetic field vector
stemming from the Maxwell equations. A key ingredient in the proof of the
improved Strichartz inequality is the Bourgain-Brezis borderline Sobolev
inequalities.
