Very weak solutions to the Navier–Stokes system in general unbounded domains

Abstract

We consider very weak instationary solutions u of the Navier–Stokes system in general unbounded domains \({\Omega \subset \mathbb{R}^n}\) , \({n \geq 3}\) , with smooth boundary, i.e., u solves the Navier–Stokes system in the sense of distributions and \({ u \in L^r (0,T;\tilde{L}^q(\Omega))}\) where \({\frac{2}{r} + \frac{n}{q} =1}\) , 2 < r < ∞. Solutions of this class have no differentiability properties and in general are not weak solutions in the sense of Leray–Hopf. However, they lie in the so-called Serrin class \({L^r(0,T;\tilde{L}^q(\Omega))}\) yielding uniqueness. To deal with the unboundedness of the domain, we work in the spaces \({\tilde{L}^q(\Omega)}\) (instead of \({L^q(\Omega)}\)) defined as \({L^q \cap L^2}\) when \({q \geq 2}\) but as Lq+ L2 when 1 < q < 2. The proofs are strongly based on duality arguments and the properties of the spaces \({\tilde{L}^q(\Omega)}\) .