If $v$ is an incompressible, irrotational fluid satisfying Euler's equation, show $\overline{v}$ is holomorphic

Question

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.

Answer 1

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.