The divergence equation in rough spaces

Raphaël Danchin (1) , Piotr Boguslaw Mucha

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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)(Omega) with s epsilon (-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 Bogovskii 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. (C) 2011 Elsevier Inc. All rights reserved.

Domaines: Physique mathématique [math-ph]

Raphaël Danchin, Piotr Boguslaw Mucha. The divergence equation in rough spaces. Journal of Mathematical Analysis and Applications, 2012, 386 (1), pp.10--31. ⟨10.1016/j.jmaa.2011.07.036⟩. ⟨hal-00692682⟩