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

Let \begin{document}$ {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $\end{document} denote a BMO space on \mathbb{R}^{n} associated to Neumann operator \mathcal{L}: = -\Delta_{N} . In this article we will show that function f\in is the trace of solution {\mathbb L}u u_{t}+ \mathcal{L} u 0, u(x, 0) f(x), where satisfies 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 t) |^2+ | \nabla_x |^2 \right\}{dx dt } \leq C for some constant C>0 Conversely, Carleson characterizes all L} -carolic functions whose traces belong Furthermore, based characterization mentioned above, prove global well-posedness parabolic equations Navier-Stokes type with boundary under smallness intial data u_{0}\in {{\rm BMO}_{\Delta_{N}}^{-1}(\mathbb{R}^{n})}