  We derive an ultraweak variational formulation of the quad-curl problem in
two and three dimensions. We present a discontinuous Petrov-Galerkin (DPG)
method for its approximation and prove its quasi-optimal convergence. We
illustrate how this method can be applied to the Stokes problem in two
dimensions, after an application of the curl operator to eliminate the pressure
variable. In this way, DPG techniques known from Kirchhoff-Love plates can be
used. We present an a priori error estimate that improves a previous
approximation result for effective shear forces by using a less restrictive
regularity assumption. Numerical experiments illustrate our findings.
