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 L ^q+ L ² when 1 < q < 2. The proofs are strongly based on duality arguments and the properties of the spaces ${\tilde{L}^q(\Omega)}$ .