Under (α,p,n-1)∈(-∞,1)×(2,∞)×N\documentclass[12pt]{minimal}
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\begin{document}$$(\alpha ,p,n-1)\in (-\infty ,1)\times (2,\infty )\times {\mathbb {N}}$$\end{document}, this paper uses Qα(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {Q}_\alpha ({\mathbb {R}}^n)$$\end{document} and Qα-1(Rn):=div(Qα(Rn))n\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {Q}^{-1}_{\alpha }(\mathbb R^n):=\hbox {div}\big (\mathcal {Q}_{\alpha }({\mathbb {R}}^{n})\big )^n$$\end{document} (covering BMO(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{BMO}({\mathbb {R}}^n)$$\end{document} and BMO-1(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{BMO}^{-1}({\mathbb {R}}^n)$$\end{document}) where f∈Qα(Rn)⇔∫Rn|f(x)|1+|x|n+1dx<∞andsupcoordinatecubesI∬I×(0,ℓ(I))|∇et2Δf(x)|2ωαtℓ(I)tn-1dxdt12<∞\documentclass[12pt]{minimal}
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\begin{document}$$ f\in \mathcal {Q}_\alpha ({\mathbb {R}}^{n}) \Leftrightarrow \int \nolimits _{\mathbb R^n}\frac{|f(x)|}{1+|x|^{n+1}}\,\mathrm{d}x<\infty \ \ \mathrm{and} \ \ \underset{\mathrm{coordinate}\, \mathrm{cubes}\, I}{\sup }\left( \iint \nolimits _{I\times (0,\ell (I))}|\nabla e^{t^2\Delta }f(x)|^2\right. \left. \frac{\omega _{\alpha }\left( \frac{t}{\ell (I)}\right) }{t^{n-1}}{\mathrm{d}x\mathrm{d}t}\right) ^\frac{1}{2}<\infty $$\end{document} with (0,1]∋s↦ωα(s)=snasα∈(-∞,0);sn(lnes)2asα=0;sn-2αasα∈(0,1),\documentclass[12pt]{minimal}
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\begin{document}$$ (0,1]\ni s\mapsto \omega _\alpha (s)={\left\{ \begin{array}{ll} s^n\quad &{}\hbox {as}\quad \alpha \in (-\infty ,0);\\ s^n\big (\ln \frac{e}{s}\Big )^2\quad &{}\hbox {as}\quad \alpha =0;\\ s^{n-2\alpha }\quad &{}\hbox {as}\quad \alpha \in (0,1), \end{array}\right. }$$\end{document} to demonstrate that the incompressible Navier–Stokes system Δu-(u·∇)u+∇p=∂tuanddivu=0inR+1+n;u(0,x)=a(x)asx∈Rn\documentclass[12pt]{minimal}
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\begin{document}$$ {\left\{ \begin{array}{ll} \Delta u-(u \cdot \nabla ) u+\nabla \mathrm{p}=\partial _t u\ \ \mathrm{and}\ \ \hbox {div}\,u=0 &{} \text {in } {\mathbb {R}}^{1+n}_+;\\ u(0,x)=a(x) &{} \text {as } x\in {\mathbb {R}}^{n} \end{array}\right. }$$\end{document} has a unique mild solution under ‖a‖(Qα-1(Rn))n\documentclass[12pt]{minimal}
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\begin{document}$$\Vert a\Vert _{\big (\mathcal {Q}_\alpha ^{-1}({\mathbb {R}}^n)\big )^n}$$\end{document} being sufficiently small; however, its steady stateΔu-(u·∇)u+∇p=0anddivu=0inRn;u(x)→0as∞←x∈Rn\documentclass[12pt]{minimal}
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\begin{document}$$ {\left\{ \begin{array}{ll} \Delta u-(u \cdot \nabla ) u+\nabla \mathrm{p}=0\ \ \mathrm{and}\ \ \hbox {div}\,u=0 &{} \text {in } {\mathbb {R}}^{n};\\ u(x)\rightarrow 0 &{} \text {as}\ \infty \leftarrow x\in {\mathbb {R}}^{n} \end{array}\right. }$$\end{document} has only zero solution under u∈(BMO-1(Rn)∩Lp,p(n-2)2(Rn))n\documentclass[12pt]{minimal}
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\begin{document}$$u\in \big (\mathrm{BMO}^{-1}(\mathbb R^n)\cap \mathscr {L}^{p,\frac{p(n-2)}{2}}({\mathbb {R}}^n)\big )^n$$\end{document}.
