BMO type space associated with Neumann operator and application to a class of parabolic equations
BMO type space associated with Neumann operator and application to a class of parabolic equations
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与诺伊曼算子相关的BMO型空间及其在一类抛物方程中的应用
DOI：
10.3934/dcdsb.2021104
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Minghua Yang
中科院分区：
文献类型：
--
作者：
Zhang Chao;Minghua Yang
Let \begin{document}$ {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $\end{document} denote a BMO space on \begin{document}$ \mathbb{R}^{n} $\end{document} associated to a Neumann operator \begin{document}$ \mathcal{L}: = -\Delta_{N} $\end{document} . In this article we will show that a function \begin{document}$ f\in {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $\end{document} is the trace of the solution of \begin{document}$ {\mathbb L}u = u_{t}+ \mathcal{L} u = 0, u(x, 0) = f(x), $\end{document} where \begin{document}$ u $\end{document} satisfies a Carleson-type condition \begin{document}$ \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 for some constant \begin{document}$ C>0 $\end{document} . Conversely, this Carleson condition characterizes all the \begin{document}$ {\mathbb L} $\end{document} -carolic functions whose traces belong to the space \begin{document}$ {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $\end{document} . Furthermore, based on the characterization of \begin{document}$ {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $\end{document} mentioned above, we prove the global well-posedness for parabolic equations of Navier-Stokes type with the Neumann boundary condition under smallness condition on the intial data \begin{document}$ u_{0}\in {{\rm BMO}_{\Delta_{N}}^{-1}(\mathbb{R}^{n})} $\end{document} .
Let \begin{document}$ {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $\end{document} denote a BMO space on \begin{document}$ \mathbb{R}^{n} $\end{document} associated to a Neumann operator \begin{document}$ \mathcal{L}: = -\Delta_{N} $\end{document} . In this article we will show that a function \begin{document}$ f\in {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $\end{document} is the trace of the solution of \begin{document}$ {\mathbb L}u = u_{t}+ \mathcal{L} u = 0, u(x, 0) = f(x), $\end{document} where \begin{document}$ u $\end{document} satisfies a Carleson-type condition \begin{document}$ \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 for some constant \begin{document}$ C>0 $\end{document} . Conversely, this Carleson condition characterizes all the \begin{document}$ {\mathbb L} $\end{document} -carolic functions whose traces belong to the space \begin{document}$ {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $\end{document} . Furthermore, based on the characterization of \begin{document}$ {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $\end{document} mentioned above, we prove the global well-posedness for parabolic equations of Navier-Stokes type with the Neumann boundary condition under smallness condition on the intial data \begin{document}$ u_{0}\in {{\rm BMO}_{\Delta_{N}}^{-1}(\mathbb{R}^{n})} $\end{document} .
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影响因子：
3
作者：
X. Duong;Lixin Yan
通讯作者：
X. Duong;Lixin Yan
DOI：
10.1007/978-3-642-14606-0
发表时间：
2010-09
期刊：
--
影响因子：
--
作者：
Wen Yuan;W. Sickel;Dachun Yang
通讯作者：
Wen Yuan;W. Sickel;Dachun Yang
影响因子：
1.1
作者：
M. Dindoš;C. Kenig;J. Pipher
通讯作者：
M. Dindoš;C. Kenig;J. Pipher
影响因子：
2.4
作者：
Y. Du;Keyan Wang
通讯作者：
Y. Du;Keyan Wang
DOI：
--
发表时间：
2014-12
期刊：
arXiv: Analysis of PDEs
影响因子：
--
作者：
P. Auscher;D. Frey
通讯作者：
P. Auscher;D. Frey