# Minimal divergence-free projection onto unit vector field

I'm attempting to use a variational method to find a critical vector field. I want to find the local components of a given vector field, $$\bf{v}$$, that are as "close" to $$\bf{v}$$ as possible, but are also divergence-free. My thought is that I can search for a vector field of unit vectors, $$\bf{u}$$, such that the projection of $$\bf{v}$$ onto $$\bf{u}$$ is divergence free and the error between this projection and $$\bf{v}$$ is minimal. I can write this as a variational problem:

$$\text{min}_{\textbf{u}} \int_V || \textbf{v} - (\textbf{v} \cdot \textbf{u})\textbf{u} || ^2 dV$$ $$\nabla \cdot (\textbf{v} \cdot \textbf{u}) \textbf{u} = 0$$ $$\textbf{u} \cdot \textbf{u} = 1$$

where $$V$$ is a smooth 2-D manifold. I can use Lagrange multipliers to remove the constraints and apply the divergence theorem to give the unconstrained variational problem:

$$\text{min}_{\textbf{u}} \int_V || \textbf{v} - (\textbf{v} \cdot \textbf{u})\textbf{u} || ^2 - \nabla \lambda_1 \cdot \textbf{u} + \lambda_2 (\textbf{u} \cdot \textbf{u} - 1) dV$$ $$\lambda_1 = 0 \quad \text{on } \partial V$$

At this point I'm not sure how to proceed. Deriving the Euler-Lagrange equations by setting $$\textbf{u} = \textbf{u}^* + \epsilon \textbf{w}$$ and setting the derivative with respect to $$\epsilon$$ to 0 at $$\epsilon = 0$$ gives a mess. I'm curious if anyone has seen a similar problem and what strategies can be used to solve this.

Thank you!

• What is $V$ here? A manifold? A disk in Euclidean space? Just a letter you pulled out of the air? :) – John Hughes Jun 18 '19 at 0:14
• This looks contrived. Why not "Find div-free w with minimal distance to v"? – user66081 Jun 18 '19 at 0:15
• @JohnHughes Sorry about that. $V$ is a 2D smooth manifold for this problem. I'll add include this in the question. – pmat Jun 18 '19 at 0:18
• @user66081 The issue with "Find div-free w with minimal distance to v" is that "w" really needs to be a component of "v" at each point on the manifold (e.g. "w" can never be bigger than "v" at any given point). I'm not sure that the nearest div-free w must have this property (could be wrong though). – pmat Jun 18 '19 at 0:22