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

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})} $.