Thinking more about these conservation laws and the connection with the nonlocal formulation, consider a harmonic function in conjunction with the velocity potential and the stream-function .
It is straightforward to show that are indeed conserved densities in the sense of Olver, 1983. But do these make sense? Well this was an embarrassing mistake, but the formulation is still helpful! Details follow below.
The Conservation Law Formulation
Ignoring total derivatives and the likes, we can write the nonlocal formulation as
Should we choose , then we find that the above reduces to
It’s critical to note that the above form can be expressed as
where and following the notation of Olver, 1983.
It is straight forward to show that the right-hand side does indeed create a closed differential and thus, the integral along the surface can be transformed to an integral along the fixed boundaries. This seems to imply that all of the functions are indeed conserved densities in the sense of Olver. Well this is where I made that sign mistake! But, it is still worth following.
Of course, two major questions arise from this. First, it’s clear that this is not a direct route to the conserved quantities of Olver. For example, conserved densities that include terms such as will not come from this form directly. However, these can be found via a suggestion in made in the post Relating the conservation laws to the Bernoulli Equation.
That said, we can still find the conservation laws that include terms of the form and the likes. For example, if we start with , we find
where we’ve used some integration-by-parts. Noting that , two of the terms are total derivatives with respect to . The remaining terms can be dealt with by playing the game with the terms to rewrite them as
Then, we only need to deal with the term term since we know that is a conserved density ( in the notation of Olver or see Relating …). Differentiating with respect to , we find which will ultimately yield a closed differential minus a contribution from gravity. That is, we find
The first integral yields a closed differential and can be replaced with integrals along the fixed boundary. The second integral can be manipulated into a closed form by making a new contribution of the form .
Combining all of this information together yields a conserved density of the form
Taking the real and imaginary parts of the above integral yields and from Olver’s hierarchy.
To find , simply take . Finding requires a different perspective as discussed in Relating ….
Thus far, I have been able to recreate and .