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 $$ |\partial_t^m\partial_x^kG_{ij}(x,y,t)|\le C_{m,k}t^{-\frac{d}{2}-m-\frac{k}{2}}e^{-\frac{|x-y|^2}{t}} . $$


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