I was wondering if there is a vector field that satisfies the following condition:
I’d like to offer a high-brow answer to this question. While I realize that this solution is likely to be above the OP’s level, I’m hoping it’ll be of some interest to others.
Claim: Let u(x,y), v(x,y), w(x) be any real-analytic functions (defined on a neighborhood of 0). Then there exists a unique real-analytic vector field F=(F1,F2,F3) that satisfies curlF=F and
We’ll show this by applying the Cauchy-Kovalevskaya Theorem twice. Here I should acknowledge that I first saw this techinque applied to curlF=F in an exercise in Robert Bryant’s “Nine Lectures on Exterior Differential Systems.”
Cauchy-Kovalevskaya Theorem: If H and ϕ are real-analytic functions near the origin, then there is a neighborhood of the origin on which there exists a unique real-analytic solution g(x,t) to
A PDE system of this form will be called a “Cauchy problem.”
The equation curlF=F can be viewed as an overdetermined system of PDEs — in fact, a system of 4 first-order quasilinear equations for 3 unknown functions. That is, if we write F=(F1,F2,F3), then the condition curlF=F becomes:
The fourth equation is a hidden “integrability condition” of sorts: namely, any solution F to curlF=F must satisfy divF=0. This gives us a fourth (first-order quasilinear) equation
The Cauchy Problems
We can write this system as a sequence of two Cauchy problems as follows. Let u(x,y), v(x,y), and w(x) be arbitrary real-analytic functions.
We first consider the problem of finding g(x,y) such that
By Cauchy-Kovalevskaya†, there exists a unique real-analytic solution g(x,y).
Second, letting g(x,y) be a solution as above, we consider the problem of finding F1,F2,F3 such that
By Cauchy-Kovalevskaya again, there exists a unique real-analytic solution F=(F1,F2,F3).
By construction, this solution (F1,F2,F3) satisfies equations (1), (2) and (4). One can check (exercise!∗) that this F necessarily satisfies equation (3) as well. This completes the proof. ◊
† Admittedly overkill here
∗ Hint for Exercise: Consider E(x,y,z):=∂F1∂y−∂F2∂x−F3. Check that E(x,y,0)=0 and ∂E∂z=0, and then apply uniqueness in Cauchy-Kovalevskaya.
Note that this exercise shows why we needed the first Cauchy problem at all: that is, we couldn’t just choose F1(x,y,0)=g(x,y) completely arbitrarily!