# Birkhoff-Rott equation

A planar vortex sheet is a curve in a two-dimensional inviscid incompressible flow across which the tangential velocity is discontinuous (cf. also Von Kármán vortex shedding). The vortex sheet is described by its complex position $z ( \Gamma , t ) = x + i y$. For simplicity, assume that the vorticity on the sheet is all positive and that the flow outside the sheet is irrotational. The sheet is parameterized by a real variable $\Gamma$ which represents the circulation, i.e. $\gamma = | \partial z / \partial \Gamma | ^ { - 1 }$ is the vorticity density along the sheet. Vortex sheet evolution is then described by the Birkhoff–Rott equation [a1], [a11]:

\begin{equation} \tag{a1} \partial _ { t } \overline{z( \Gamma , t )} = ( 2 \pi i ) ^ { - 1 } \operatorname{PV} \int _ { - \infty } ^ { \infty } \frac { d \Gamma ^ { \prime } } { z ( \Gamma , t ) - z ( \Gamma ^ { \prime } , t ) }. \end{equation}

Because of the singularity of the integral at $\Gamma ^ { \prime } = \Gamma$, the integral in (a1) is understood as a Cauchy principal value integral (cf. also Cauchy integral).

Perturbations of a flat sheet of uniform strength grow due to the linear Kelvin–Helmholtz instability and at some time later the sheet begins to roll-up. D. Moore [a8], [a9] showed by asymptotic analysis that a singularity could develop along the sheet at finite time starting from smooth initial data. The singularity found by Moore has the form $z _ { \Gamma } = \mathcal{O} ( \Gamma ^ { - 1 / 2 } )$ in which $z ( \Gamma ) = x + i y$ is the position and $\Gamma$ is the circulation variable. This singularity form was later found to be generic [a16]. Exact singular solutions of the non-linear Birkhoff–Rott equation, corresponding to Moore's singularity, have been constructed in [a3], [a4].

Numerical simulations of the vortex sheet problem [a5], [a7], [a12] have produced singular solutions which are in agreement with Moore's theory. Krasny's method [a5] used a non-linear filter to remove the numerical noise generated by the physical instability, the convergence of which was proved in [a15] for analytic initial data. R. Krasny [a6] also computed roll-up of a sheet, using a desingularized equation, and found that the sheet begins to roll-up immediately after the appearance of the first singularity. A general set of similarity solutions for a rolled-up vortex sheet were constructed numerically in [a10].

Existence results almost up to the singularity time have been proved [a2], [a13], using the abstract Cauchy–Kovalevskaya theorem. The results for existence and for singularity formation use an extension of the Birkhoff–Rott equation (a1) into the complex $\Gamma$-plane for analytic initial data. Since the linearization of (a1) is elliptic in $\Gamma$ and $t$ (cf. also Elliptic partial differential equation), it is hyperbolic in the imaginary $\Gamma$ direction (cf. also Hyperbolic partial differential equation). Singularities in the initial data at complex values of $\Gamma$ travel towards the real axis at a finite speed.

The Birkhoff–Rott equation has been extended to three-dimensional sheets in [a14]. Short-time existence theory for the three-dimensional equations has been established in [a13]. A computational method for the three-dimensional equations was implemented in [a17].

Open questions as of 2000 include the well-posedness for continuation after Moore's singularity and the form of singularities in three dimensions.

How to Cite This Entry:
Birkhoff-Rott equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birkhoff-Rott_equation&oldid=50380
This article was adapted from an original article by Russel E. Caflisch (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article