# Existence/Uniqueness for Stokes flow with pressure boundary conditions

We consider here the (2D) Stokes flow equations \begin{align} - \nabla^2 \mathbf{u} + \nabla p = 0 \\ \nabla \cdot \mathbf{u} = 0 \end{align} where we have set the viscosity $$\nu = 1$$ for simplicity. I want to understand the existence/uniqueness of this problem in a bounded domain $$\Omega$$.

In the math literature (e.g. this answer or this paper or this note) one often works with fixed velocity boundary conditions (i.e. $$\mathbf{u} \in H_0^1 (\Omega)$$). However, I am interested in the case where one imposes fixed pressure on segments of $$\partial \Omega$$.

To be concrete, let me take $$\Omega = [0,1]\times[0,1]$$, where $$p(0,y) = p(1,y) = 0$$ and $$\mathbf{u}(x,0) = \mathbf{u}(x,1)=0$$. In other words, take a rectangular box where the pressure is fixed to zero on the left and right boundaries, and take a no-slip condition on the top and bottom boundaries. This is a typical physical setup for pipe flow, where one specifies the pressure rather than the velocities on the inlet/outlets of the pipe. Are there existence/uniqueness results for this case?

• The incompressible NSE (and also the stokes equations) are first order in the variable $p$, so using a Dirichlet boundary condition on two of the faces usually gives bad results, in my experience. Sep 10 at 2:06