Existence/Uniqueness for Stokes flow with pressure boundary conditions

Question

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?

Answer 1

Unfortunately you don't get to specify only pressure (one scalar function) on the boundary; that is not enough data. You only get a well-posed interior BVP if you specific traction, ie, the stress tensor dotted into the boundary normal. This is a vector function, interpreted as the force density (including pressure and viscous drag) on the fluid at each boundary point. It is the Stokes equivalent of Neumann data for the scalar Laplace BVP. The pure interior traction BVP has a solution iff the traction data obeys zero net force and torque. The solution is unique up to rigid-body motions in u. This is summarized in the Hsiao & Wendland book, Eqns (2.3.24) and (2.3.25), p.66, but it's a technical book, and boundary integral operators are needed for the proofs.

The classical work is Ladyzhenskaya's 1969 book, see p.59-60, that you may find clearer and less technical.

Now, you describe a mixed boundary condition (velocity on part of the boundary, and it would have to be traction as opposed to pressure on the remaining part). Of the top of my head I don't know the compatibility conditions for well-posedness, but there may not be any.

Searching for Stokes (as opposed to Navier--Stokes) results in the literature is hard since most is about N--S. However, since Hsiao--W book shows the Stokes equations are a special case of incompressible elasticity, for which there are many papers about the well-posedness of the traction problem, that might be a better route to go for what you need. Good luck!