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BMO type space associated with Neumann operator and application to a class of parabolic equations

Zhang Chao's work was supported by National Natural Science Foundation of China(Grant Nos. 11971431, 11401525), the Natural Science Foundation of Zhejiang Province (Grant No. LY18A010006) and the first Class Discipline of Zhejiang-A(Zhejiang Gongshang University- Statistics). Minghua Yang's work was supported by the National Natural Science Foundation of China (Grant No. 11801236), Postdoctoral Science Foundation of China (Grant Nos. 2020T130265, 2018M632593), Natural Science Foundation of Jiangxi Province (Grant No.20204BCJL23056), the Postdoctoral Science Foundation of Jiangxi Province (Grant No. 2017KY23) and Educational Commission Science Programm of Jiangxi Province (Grant No. GJJ190272)

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  • Let $ {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $ denote a BMO space on $ \mathbb{R}^{n} $ associated to a Neumann operator $ \mathcal{L}: = -\Delta_{N} $. In this article we will show that a function $ f\in {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $ is the trace of the solution of $ {\mathbb L}u = u_{t}+ \mathcal{L} u = 0, u(x, 0) = f(x), $ where $ u $ satisfies a Carleson-type condition

    $ \begin{eqnarray*} \sup\limits_{x_B, r_B} r_B^{-n}\int_0^{r_B^2}\int_{B(x_B, r_B)} \left\{t| \partial_t u(x, t) |^2+ | \nabla_x u(x, t) |^2 \right\}{dx dt } \leq C <\infty, \end{eqnarray*} $

    for some constant $ C>0 $. Conversely, this Carleson condition characterizes all the $ {\mathbb L} $-carolic functions whose traces belong to the space $ {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $. Furthermore, based on the characterization of $ {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $ mentioned above, we prove the global well-posedness for parabolic equations of Navier-Stokes type with the Neumann boundary condition under smallness condition on the intial data $ u_{0}\in {{\rm BMO}_{\Delta_{N}}^{-1}(\mathbb{R}^{n})} $.

    Mathematics Subject Classification: Primary: 42B35, 47F05.


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