Rough solutions for the periodic Korteweg-de Vries equation

We show how to apply ideas from the theory of rough paths to the analysis of low-regularity solutions to non-linear dispersive equations. Our basic example will be the one dimensional Korteweg--de Vries (KdV) equation on a periodic domain and with initial condition in $\FF L^{\alpha,p}$ spaces. We discuss convergence of Galerkin approximations, a modified Euler scheme and the presence of a random force of white-noise type in time.