The coupling between dilatation and vorticity, two coexisting and fundamental
processes in fluid dynamics is investigated here, in the simplest cases of
inviscid 2D isotropic Burgers and pressureless Euler-Coriolis fluids
respectively modeled by single vortices confined in compressible, local,
inertial and global, rotating, environments. The field equations are
established, inductively, starting from the equations of the characteristics
solved with an initial Helmholtz decomposition of the velocity fields namely a
vorticity free and a divergence free part and, deductively, by means of a
canonical Hamiltonian Clebsch like formalism, implying two pairs of conjugate
variables. Two vector valued fields are constants of the motion: the velocity
field in the Burgers case and the momentum field per unit mass in the
Euler-Coriolis one. Taking advantage of this property, a class of solutions for
the mass densities of the fluids is given by the Jacobian of their sum with
respect to the actual coordinates. Implementation of the isotropy hypothesis
results in the cancellation of the dilatation-rotational cross terms in the
Jacobian. A simple expression is obtained for all the radially symmetric
Jacobians occurring in the theory. Representative examples of regular and
singular solutions are shown and the competition between dilatation and
vorticity is illustrated. Inspired by thermodynamical, mean field theoretical
analogies, a genuine variational formula is proposed which yields unique
measure solutions for the radially symmetric fluid densities investigated. We
stress that this variational formula, unlike the Hopf-Lax formula, enables us
to treat systems which are both compressible and rotational. Moreover in the
one-dimensional case, we show for an interesting application that both
variational formulas are equivalent
