It is well known that except Poisson problem also the Dirichlet problem
$$\Delta u = 0, \qquad u = u_0 \; \text{at}\; \partial B_1(0)$$
possesses in 3D (actually in any dimension) an exact (in a sense of one integral) solution
$$u(\vec{x}) = \frac{1}{4\pi} \oint_{B_R(0)} u_0(\vec{x}') \frac{R^2-|\vec{x}|^2}{|\vec{x}-\vec{x}'|^2}dS'$$
Similarly, decomposition into components yields a formula for the vector Dirichlet problem
$$\Delta \vec{A} = 0, \qquad \vec{A} = \vec{A}_0 \; \text{at}\; \partial B_R(0)$$
has a solution
$$\vec{A}(\vec{x}) = \frac{1}{4\pi} \oint_{B_R(0)} \vec{A}_0(\vec{x}') \frac{R^2-|\vec{x}|^2}{|\vec{x}-\vec{x}'|^3}dS'$$
My question is, I am solving a different problem (hydrodynamics) namely a system of the Stokes flow equations :
$$\nabla p = \Delta\vec{v},\qquad \nabla\cdot\vec{v}=0$$
After taking a curl clearly $\Delta p=0$. Let us suppose $p$ vanishes at infinity. Given a boundary condition
$$\vec{v} = \vec{v}_0 \; \text{at}\; \partial B_R(0)$$
is there a similar general solution for $\vec{v}$ (and $p$) using nothing more than an appropriate Green function?