If $v$ is an incompressible, irrotational fluid satisfying Euler's equation, show $\overline{v}$ is holomorphic This feels like it'll be simple once I see it, but I've been stuck for a while now. The setup is from Sijue Wu's 1997 paper Well-posedness in Sobolev spaces of the full water wave problem in 2D. $v=(v_1,v_2)$ is the fluid velocity, $p$ is the pressure, and $\Omega(t)$ is essentially the water region. Then
$$
\begin{cases}
v_t + v\cdot \nabla v & = -(0,1)-\nabla p\\
\operatorname{div}(v) & = 0\\
\operatorname{curl}(v) & = 0
\end{cases}
$$Viewing $z=(x,y)$ as a complex number, if we write $z$ in Lagrangian coordinates with parameter $\alpha$
$$
z_t(\alpha,t) = v(z(\alpha,t),t),
$$the claim is that the last two equations imply $\overline{v}(\alpha,t) = v_1(\alpha,t)-i v_2 (\alpha,t)$ is holomorphic. I had thought to use the Cauchy-Riemann equations to try to show
$$
\frac{\partial v_1}{\partial \alpha} = -\frac{\partial v_2}{\partial t};\qquad \frac{\partial v_1}{\partial t} = -\frac{\partial v_2}{\partial \alpha}
$$but I'm embarassed to say I got mixed up doing the Chain Rule. Am I on the right track? I feel like I'm  not fully using the information about being incompressible and irrotational, particular the curl equation as it should relate the partial derivatives.
 A: You have a 2D flow that is incompressible and irrotational.  This implies that the velocity components everywhere satisfy
$$\tag{*}\nabla \cdot \mathbf{v} = 0 \implies\frac{\partial v_x}{\partial x}= - \frac{\partial v_y}{\partial y}, \quad \\ \nabla \times \mathbf{v} = 0 \implies \frac{\partial v_x}{\partial y}=  \frac{\partial v_y}{\partial x}$$
Also $\mathbb{v} = \nabla \phi$, the gradient of a potential, and there exists a streamfunction $\psi$ such that the complex potential $f(z) = \phi + i \psi$ is holomorphic.  By the Cauchy-Riemann equations, we have
$$v_x = \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}, \quad v_y = \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x} $$
The complex derivative of $f$ is the complex velocity given by
$$\frac{df}{dz}= \frac{\partial \phi}{\partial x}+ i \frac{\partial \psi}{\partial x}= v_x - i v_y$$
Since $f$ is everywhere holomorphic, so is $\frac{df}{dz}$.  This also can be seen from condition (*) which shows that the real and imaginary parts of the complex velocity also satisfy the Cauchy-Riemann equations.
