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

Zhang Chao; Minghua Yang

doi:10.3934/dcdsb.2021104

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March 2022, 27(3): 1629-1645. doi: 10.3934/dcdsb.2021104

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

Zhang Chao ^1, and Minghua Yang ^2,,

1.	School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China
2.	Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang 330032, China

^*Corresponding author: yangmh7@mail2.sysu.edu.cn

Received November 2020 Revised January 2021 Published March 2022 Early access March 2021

Fund Project: 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)

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})} $

Keywords: Neumann operator, BMO space, Carleson measure, parabolic equation.

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

Citation: Zhang Chao, Minghua Yang. BMO type space associated with Neumann operator and application to a class of parabolic equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1629-1645. doi: 10.3934/dcdsb.2021104