We consider very weak instationary solutions u of the Navier–Stokes system in general unbounded domains Ω⊂Rn\documentclass[12pt]{minimal}
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\begin{document}$${\Omega \subset \mathbb{R}^n}$$\end{document} , n≥3\documentclass[12pt]{minimal}
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\begin{document}$${n \geq 3}$$\end{document} , with smooth boundary, i.e., u solves the Navier–Stokes system in the sense of distributions and u∈Lr(0,T;L~q(Ω))\documentclass[12pt]{minimal}
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\begin{document}$${ u \in L^r (0,T;\tilde{L}^q(\Omega))}$$\end{document} where 2r+nq=1\documentclass[12pt]{minimal}
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\begin{document}$${\frac{2}{r} + \frac{n}{q} =1}$$\end{document} , 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 Lr(0,T;L~q(Ω))\documentclass[12pt]{minimal}
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\begin{document}$${L^r(0,T;\tilde{L}^q(\Omega))}$$\end{document} yielding uniqueness. To deal with the unboundedness of the domain, we work in the spaces L~q(Ω)\documentclass[12pt]{minimal}
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\begin{document}$${\tilde{L}^q(\Omega)}$$\end{document} (instead of Lq(Ω)\documentclass[12pt]{minimal}
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\begin{document}$${L^q(\Omega)}$$\end{document}) defined as Lq∩L2\documentclass[12pt]{minimal}
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\begin{document}$${L^q \cap L^2}$$\end{document} when q≥2\documentclass[12pt]{minimal}
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\begin{document}$${q \geq 2}$$\end{document} but as Lq+ L2 when 1 < q < 2. The proofs are strongly based on duality arguments and the properties of the spaces L~q(Ω)\documentclass[12pt]{minimal}
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\begin{document}$${\tilde{L}^q(\Omega)}$$\end{document} .
