  We prove that in a certain asymptotic regime, solutions of the
Gross-Pitaevskii equation converge to solutions of the incompressible Euler
equation, and solutions to the parabolic Ginzburg-Landau equations converge to
solutions of a limiting equation which we identify.
  We work in the setting of the whole plane (and possibly the whole
three-dimensional space in the Gross-Pitaevskii case), in the asymptotic limit
where $\epsilon$, the characteristic lengthscale of the vortices, tends to $0$,
and in a situation where the number of vortices $N$ blows up as $\epsilon \to
0$. The requirements are that $N$ should blow up faster than $|\log \epsilon|$
in the Gross-Pitaevskii case, and at most like $|\log \epsilon|$ in the
parabolic case. Both results assume a well-prepared initial condition and
regularity of the limiting initial data, and use the regularity of the solution
to the limiting equation.
  In the case of the parabolic Ginzburg-Landau equation, the limiting
mean-field dynamical law that we identify coincides with the one proposed by
Chapman-Rubinstein-Schatzman and E in the regime $N\ll |\log \epsilon|$, but
not if $N$ grows faster.
