Inverting the vorticity via Biot-Savart in Navier Stokes Given the Navier Stokes equation $\partial_t u+u\cdot \nabla u+\nabla p=\nu \Delta u$ in $\mathbb{R}^3$ with $u$ divergence-free, one is often interested in the vorticity $\omega=\text{curl} \ u$. In Majda Bertozzi (Chapter 2.4, page 73) it is claimed that one has the Biot-Savart Law
$$u(t,x)=\dfrac{1}{4\pi} \int_{\mathbb{R}^3 } \dfrac{(x-y) \times \omega (t,y)}{|x-y|^3} dy $$
To my understanding, they way they arrive at this solution is by showing that the system $\text{curl} \ u =\omega,\ \text{div}\ u=0$ has the above solution with which I agree (they don't do brute force computation, they take $\psi$ such that $\Delta \psi=\omega$ and it turns out that $-\text{curl} \ \psi$ has the desired properties and is of the above form).
What is not clear to me is why $u$ should be exactly that;  the system $\text{curl} \ u =\omega,\ \text{div}\ u=0$ clearly does not have a unique solution since the system $\text{curl} \ u =0,\ \text{div}\ u=0$ does not have a unique solution. Should there not be an additional freedom given by some $\nabla f$ where $f:\mathbb{R}^3\rightarrow \mathbb{R}$ is harmonic? Freedom in the sense that
$$u(x)=\dfrac{1}{4\pi} \int_{\mathbb{R}^3 } \dfrac{(x-y) \times \omega (t,y)}{|x-y|^3} dy\ + \ \nabla f $$
This would make intuitive sense since $\omega$ depends only on the derivatives of $u$ so it "loses" information.
I've seen the formula from Majda Bertozzi even in papers and it's not clear to me why nobody talks about the extra term $\nabla f$. If anybody has any input I'd appreciate it.
 A: The Biot-Savart law only determines the velocity field up to an arbitrary irrotational component $\nabla f$.  Since $\nabla \times \nabla f= 0$, the presence of this term does not change the vorticity.
The function $f$ is required to enforce a no-flux condition at a bounding surface $S$.  For incompressible flow, we have the divergence-free condition $\nabla \cdot \nabla f = \nabla^2f= 0$, and $f$ can be determined uniquely as the solution to the potential problem
$$\nabla^2 f = 0, \quad \frac{\partial f}{\partial n} + \mathbf{u}_\omega\cdot \mathbf{n}= U_n \,\text{ for } \mathbf{x} \in S. \quad f(\mathbf{x}) \to 0 \, \text{ as } \, \mathbf{x} \to \infty ,$$
where $\mathbf{u}_\omega$ is the Biot-Savart velocity field and $U_n$ is a specified surface velocity (zero for no flux through a stationary surface).
A: The Helmholtz (-Hodge) decomposition theorem in 3D states that any vector F can be decomposed into a curl-free and divergence-free component: F=-∇Φ+∇×A.
See e.g. https://en.wikipedia.org/wiki/Helmholtz_decomposition.
Majda & Bertozzi deal with incompressible flow, and, therefore, Φ=0!
By the way, the formula you refer to lacks a minus sign (see e.g. "An Introduction to Theoretical Fluid Dynamics", Stephen Childress, February 12, 2008).
