View source for 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|Von Kármán vortex shedding]]). The vortex sheet is described by its complex position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130150/b1301501.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130150/b1301502.png" /> which represents the circulation, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130150/b1301503.png" /> is the vorticity density along the sheet. Vortex sheet evolution is then described by the Birkhoff–Rott equation [[#References|[a1]]], [[#References|[a11]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130150/b1301504.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

Because of the singularity of the integral at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130150/b1301505.png" />, the integral in (a1) is understood as a Cauchy principal value integral (cf. also [[Cauchy integral|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 [[#References|[a8]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130150/b1301506.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130150/b1301507.png" /> is the position and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130150/b1301508.png" /> is the circulation variable. This singularity form was later found to be generic [[#References|[a16]]]. Exact singular solutions of the non-linear Birkhoff–Rott equation, corresponding to Moore's singularity, have been constructed in [[#References|[a3]]], [[#References|[a4]]].

Numerical simulations of the vortex sheet problem [[#References|[a5]]], [[#References|[a7]]], [[#References|[a12]]] have produced singular solutions which are in agreement with Moore's theory. Krasny's method [[#References|[a5]]] used a non-linear filter to remove the numerical noise generated by the physical instability, the convergence of which was proved in [[#References|[a15]]] for analytic initial data. R. Krasny [[#References|[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 [[#References|[a10]]].

Existence results almost up to the singularity time have been proved [[#References|[a2]]], [[#References|[a13]]], using the abstract [[Cauchy–Kovalevskaya theorem|Cauchy–Kovalevskaya theorem]]. The results for existence and for singularity formation use an extension of the Birkhoff–Rott equation (a1) into the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130150/b1301509.png" />-plane for analytic initial data. Since the linearization of (a1) is elliptic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130150/b13015010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130150/b13015011.png" /> (cf. also [[Elliptic partial differential equation|Elliptic partial differential equation]]), it is hyperbolic in the imaginary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130150/b13015012.png" /> direction (cf. also [[Hyperbolic partial differential equation|Hyperbolic partial differential equation]]). Singularities in the initial data at complex values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130150/b13015013.png" /> travel towards the real axis at a finite speed.

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

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Birkhoff,   "Helmholtz and Taylor instability" , ''Proc. Symp. Appl. Math.'' , '''XII''' , Amer. Math. Soc.  (1962)  pp. 55–76</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.E. Caflisch,   O.F. Orellana,   "Long time existence for a slightly perturbed vortex sheet"  ''Commun. Pure Appl. Math.'' , '''39'''  (1986)  pp. 807–838</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.E. Caflisch,   O.F. Orellana,   "Singularity formulation and ill-posedness for vortex sheets"  ''SIAM J. Math. Anal.'' , '''20'''  (1989)  pp. 293–307</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Duchon,   R. Robert,   "Global vortex sheet solutions of Euler equations in the plane"  ''J. Diff. Eqs.'' , '''73'''  (1988)  pp. 215–224</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Krasny,   "On singularity formation in a vortex sheet and the point vortex approximation"  ''J. Fluid Mech.'' , '''167'''  (1986)  pp. 65–93</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Krasny,   "Desingularization of periodic vortex sheet roll-up"  ''J. Comput. Phys.'' , '''65'''  (1986)  pp. 292–313</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D.I. Meiron,   G.R. Baker,   S.A. Orszag,   "Analytic structure of vortex sheet dynamics, Part 1, Kelvin–Helmholtz instability"  ''J. Fluid Mech.'' , '''114'''  (1982)  pp. 283–298</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  D.W. Moore,   "The spontaneous appearance of a singularity in the shape of an evolving vortex sheet"  ''Proc. Royal Soc. London A'' , '''365'''  (1979)  pp. 105–119</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  D.W. Moore,   "Numerical and analytical aspects of Helmholtz instability"  F.I. Niordson (ed.)  N. Olhoff (ed.) , ''Theoretical and Applied Mechanics (Proc. XVI ICTAM)'' , North-Holland  (1984)  pp. 629–633</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  D.I. Pullin,   W.R.C. Phillips,   "On a generalization of Kaden's problem"  ''J. Fluid Mech.'' , '''104'''  (1981)  pp. 45–53</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  N. Rott,   "Diffraction of a weak shock with vortex generation"  ''JFM'' , '''1'''  (1956)  pp. 111</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  M. Shelley,   "A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method"  ''J. Fluid Mech.'' , '''244'''  (1992)  pp. 493–526</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  P. Sulem,   C. Sulem,   C. Bardos,   U. Frisch,   "Finite time analyticity for the two and three dimensional Kelvin–Helmoltz instability"  ''Comm. Math. Phys.'' , '''80'''  (1981)  pp. 485–516</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  R.E. Caflisch,   X. Li,   "Lagrangian theory for 3D vortex sheets with axial or helical symmetry"  ''Transport Th. Statist. Phys.'' , '''21'''  (1992)  pp. 559–578</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  R.E. Caflisch,   T.Y. Hou,   J. Lowengrub,   "Almost optimal convergence of the point vortex method for vortex sheets using numerical filtering"  ''Math. Comput.'' , '''68'''  (1999)  pp. 1465–1496</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  S.J. Cowley,   G.R. Baker,   S. Tanveer,   "On the formation of Moore curvature singularities in vortex sheets"  ''J. Fluid Mech.'' , '''378'''  (1999)  pp. 233–267</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  M. Brady,   A. Leonard,   D.I. Pullin,   "Regularized vortex sheet evolution in three dimensions"  ''J. Comput. Phys.'' , '''146'''  (1998)  pp. 520–45</TD></TR></table>