The divergence equation in rough spaces

We aim at extending the existence theory for the equation div v = f in a bounded or exterior domain with homogeneous Dirichlet boundary conditions, to a class of solutions which need not have a trace at the boundary. Typically, the weak solutions that we shall consider will belong to some Besov space B p , q s ( Ω ) with s ∈ ( − 1 + 1 / p , 1 / p ) . After generalizing the notion of a solution for this equation, we propose an explicit construction by means of the classical Bogovskiĭ formula. This construction enables us to keep track of a “marginal” information about the trace of solutions. In particular, it ensures that the trace is zero if f is smooth enough. We expect our approach to be of interest for the study of rough solutions to systems of fluid mechanics.