# Green's function of time-dependent Stokes equation

It is well known that the Green's function of a standard parabolic equation in a bounded domain, say \begin{align} \partial_tG(x,y,t)-\Delta_x G(x,y,t)&=0 &&\mbox{for}\,\,\,(x,t)\in \Omega\times{\mathbb R}_+ , \\ G(x,y,t)&=0 &&\mbox{for}\,\,\,(x,t)\in\partial\Omega\times{\mathbb R}_+ ,\\ G(x,y,0)&=\delta(x-y) &&\mbox{for}\,\,\,x\in \Omega , \end{align} with $$y\in\Omega$$, satisfies the following pointwise Gaussian estimate: $$|G(x,y,t)|\le Ct^{-\frac{d}{2}}e^{-\frac{|x-y|^2}{t}} .$$ Higher-order partial derivatives of Green's function satisfy $$|\partial_t^m\partial_x^kG(x,y,t)|\le C_{m,k}t^{-\frac{d}{2}-m-\frac{k}{2}}e^{-\frac{|x-y|^2}{t}} .$$

Is there a similar Gaussian estimate for Green's function of the time-dependent Stokes equation ? The Green function of the time-dependent Stokes equation is a matrix $$G_{ij}(x,y,t)$$, satisfying the following equation: \begin{align} \partial_t G_{ij}(x,y,t)-\Delta_x G_{ij}(x,y,t) + \partial_j P_{i}(x,y,t) &=0 &&\mbox{for}\,\,\,(x,t)\in \Omega\times{\mathbb R}_+ ,\\ \sum_{j=1}^d\partial_j G_{ij}(x,y,t)&=0 &&\mbox{for}\,\,\,(x,t)\in \Omega\times{\mathbb R}_+ ,\\ G_{ij}(x,y,t)&=0 &&\mbox{for}\,\,\,(x,t)\in\partial\Omega\times{\mathbb R}_+ ,\\ G_{ij}(x,y,0)&=\delta_{ij}(x-y) &&\mbox{for}\,\,\,x\in \Omega , \end{align} where \delta_{ij}(x-y)=\left\{\begin{aligned} &\delta(x-y) &&\mbox{if}\,\,\,i=j,\\ &0 &&\mbox{if}\,\,\,i\neq j. \end{aligned}\right.

Equivalently, if we denote $${\bf G}_i(x,y,t)$$ to be the $$i$$th row of the matrix $$G_{ij}(x,y,t)$$, then
\begin{align} \partial_t {\bf G}_{i}(x,y,t)-\Delta_x {\bf G}_{i}(x,y,t) + \nabla P_{i}(x,y,t) &=0 &&\mbox{for}\,\,\,(x,t)\in \Omega\times{\mathbb R}_+ , \\ \nabla\cdot {\bf G}_{i}(x,y,t)&=0 &&\mbox{for}\,\,\,(x,t)\in \Omega\times{\mathbb R}_+ , \\ {\bf G}_{i}(x,y,t)&=0 &&\mbox{for}\,\,\,(x,t)\in\partial\Omega\times{\mathbb R}_+ ,\\ {\bf G}_{i}(x,y,0)&=(\delta_{i1},\delta_{i2},\delta_{i3})\,\delta(x-y) &&\mbox{for}\,\,\,x\in \Omega . \end{align}

The question is whether the following estimate holds ? $$|G_{ij}(x,y,t)|\le Ct^{-\frac{d}{2}}e^{-\frac{|x-y|^2}{t}} ,$$ and $$|\partial_t^m\partial_x^kG_{ij}(x,y,t)|\le C_{m,k}t^{-\frac{d}{2}-m-\frac{k}{2}}e^{-\frac{|x-y|^2}{t}} .$$