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N–S Systems via \(\mathcal {Q}\)–\(\mathcal {Q}^{-1}\) Spaces

Research paper by Jie Xiao, Junjie Zhang

Indexed on: 15 Jun '18Published on: 14 Jun '18Published in: The Journal of Geometric Analysis



Abstract

Under \((\alpha ,p,n-1)\in (-\infty ,1)\times (2,\infty )\times {\mathbb {N}}\), this paper uses \(\mathcal {Q}_\alpha ({\mathbb {R}}^n)\) and \(\mathcal {Q}^{-1}_{\alpha }(\mathbb R^n):=\hbox {div}\big (\mathcal {Q}_{\alpha }({\mathbb {R}}^{n})\big )^n\) (covering \(\mathrm{BMO}({\mathbb {R}}^n)\) and \(\mathrm{BMO}^{-1}({\mathbb {R}}^n)\)) where \( f\in \mathcal {Q}_\alpha ({\mathbb {R}}^{n}) \Leftrightarrow \int \nolimits _{\mathbb R^n}\frac{|f(x)|}{1+|x|^{n+1}}\,\mathrm{d}x<\infty \ \ \mathrm{and} \ \ \underset{\mathrm{coordinate}\, \mathrm{cubes}\, I}{\sup }\left( \iint \nolimits _{I\times (0,\ell (I))}|\nabla e^{t^2\Delta }f(x)|^2\right. \left. \frac{\omega _{\alpha }\left( \frac{t}{\ell (I)}\right) }{t^{n-1}}{\mathrm{d}x\mathrm{d}t}\right) ^\frac{1}{2}<\infty \) with \( (0,1]\ni s\mapsto \omega _\alpha (s)={\left\{ \begin{array}{ll} s^n\quad &{}\hbox {as}\quad \alpha \in (-\infty ,0);\\ s^n\big (\ln \frac{e}{s}\Big )^2\quad &{}\hbox {as}\quad \alpha =0;\\ s^{n-2\alpha }\quad &{}\hbox {as}\quad \alpha \in (0,1), \end{array}\right. }\) to demonstrate that the incompressible Navier–Stokes system \( {\left\{ \begin{array}{ll} \Delta u-(u \cdot \nabla ) u+\nabla \mathrm{p}=\partial _t u\ \ \mathrm{and}\ \ \hbox {div}\,u=0 &{} \text {in } {\mathbb {R}}^{1+n}_+;\\ u(0,x)=a(x) &{} \text {as } x\in {\mathbb {R}}^{n} \end{array}\right. }\) has a unique mild solution under \(\Vert a\Vert _{\big (\mathcal {Q}_\alpha ^{-1}({\mathbb {R}}^n)\big )^n}\) being sufficiently small; however, its steady state\( {\left\{ \begin{array}{ll} \Delta u-(u \cdot \nabla ) u+\nabla \mathrm{p}=0\ \ \mathrm{and}\ \ \hbox {div}\,u=0 &{} \text {in } {\mathbb {R}}^{n};\\ u(x)\rightarrow 0 &{} \text {as}\ \infty \leftarrow x\in {\mathbb {R}}^{n} \end{array}\right. }\) has only zero solution under \(u\in \big (\mathrm{BMO}^{-1}(\mathbb R^n)\cap \mathscr {L}^{p,\frac{p(n-2)}{2}}({\mathbb {R}}^n)\big )^n\).