Let $L$ be a Neumann operator of the form $L=-\Delta_{ N}$ acting on $L^2(\mathbb R^n)$. Let ${ BMO}_{ \Delta_{ N}}(\mathbb R^n)$ denote the BMO space on $\mathbb R^n$ associated to the Neumann operator $\L$. In this article we will show that a function $f\in { BMO}_{ \Delta_{ N}}(\mathbb R^n)$ is the trace of the solution of $${ \mathbb L}u=u_{ t}+L u=0, u(x,0)= f(x),$$ where $u$ satisfies a Carleson-type condition \begin{ eqnarray*} \sup_{ x_B, r_B} r_B^{ -n}\int_0^{ r_B^2}\int_{ B(x_B, r_B)} |\nabla u(x,t)|^2 { 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 ${ BMO}_{ \Delta_{ N}}(\mathbb R^n)$. This result extends the analogous characterization founded by E. Fabes and U. Neri in ({ Duke Math. J.} { 42} (1975), 725-734) for the classical BMO space of John and Nirenberg. Furthermore, based on the characterization of ${ BMO}_{ \Delta_{ N}}(\mathbb R^n)$ space mentioned above, we prove global well-posedness for parabolic equations of Navier-Stokes type with the Neumann boundary condition under smallness condition on intial data $u_{ 0}\in { { BMO}_{ \Delta_{ N}}^{ -1}(\mathbb R^n)}$, which is motivated by the work of P. Auscher and D. Frey ({ J. Inst. Math. Jussieu} { 16(5)} (2017), 947-985).
Categories: math.AP
