*Published Paper*

**Inserted:** 12 dec 2002

**Journal:** ESAIM, C.O.C.V.

**Number:** 8

**Pages:** 219-238

**Year:** 2002

**Abstract:**

We consider complex-valued solutions $u_\varepsilon$ of
the Ginzburg-Landau equation on a smooth bounded simply connected
domain $\Omega$ of $*R*^N$, $N\ge 2$, where $\varepsilon>0$ is a small parameter. We assume that the
Ginzburg-Landau energy $E_\varepsilon(u_\varepsilon)$ verifies the bound (natural in the context)
$E_\varepsilon(u_\varepsilon)\le M_0

\log\varepsilon

$,
where $M_0$ is some given constant. We
also make several assumptions on the boundary data. An
important step in the asymptotic analysis of $u_\varepsilon$, as
$\varepsilon\to 0$, is to establish uniform $L^p$ bounds for the
gradient, for some $p>1$. We review some recent techniques developed in the elliptic case
in 7, discuss some variants, and extend the methods to the associated parabolic
equation.

**Keywords:**
parabolic equations, Hodge decomposition, Ginzburg-Landau, Jacobians