  We study the one-dimensional symmetry of solutions to the nonlinear Stokes
equation $$ \begin{cases} -\Delta u+\nabla W(u)=\nabla p&\text{in
}\mathbb{R}^d,\\ \nabla\cdot u=0&\text{in }\mathbb{R}^d, \end{cases} $$ which
are periodic in the $d-1$ last variables (living on the torus
$\mathbb{T}^{d-1}$) and globally minimize the corresponding energy in
$\Omega=\mathbb{R}\times \mathbb{T}^{d-1}$, i.e., $$ E(u)=\int_{\Omega} \frac12
|\nabla u|^2+W(u)\, dx, \quad \nabla\cdot u=0. $$ Namely, we determine a class
of nonlinear potentials $W\geq 0$ such that any global minimizer $u$ of $E$
connecting two zeros of $W$ as $x_1\to\pm\infty$ is one-dimensional, i.e., $u$
depends only on the $x_1$ variable. In particular, this class includes in
dimension $d=2$ the nonlinearities $W=w^2$ with $w$ being an harmonic function
or a solution to the wave equation, while in dimension $d\geq 3$, this class
contains a perturbation of the Ginzburg-Landau potential as well as potentials
$W$ having $d+1$ wells with prescribed transition cost between the wells. For
that, we develop a theory of calibrations relying on the notion of entropy
(coming from scalar conservation laws). We also study the problem of the
existence of global minimizers of $E$ for general potentials $W$ providing in
particular compactness results for uniformly finite energy maps $u$ in $\Omega$
connecting two wells of $W$ as $x_1\to\pm\infty$.
