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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]]]:
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<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>
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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]]).
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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 $z ( \Gamma , t ) = x + i y$. 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 [[#References|[a1]]], [[#References|[a11]]]:
  
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]]].
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\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}
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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|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 $z _ { \Gamma } = \mathcal{O} ( \Gamma ^ { - 1 / 2 } )$ 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 [[#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]]].
 
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.
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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 $\Gamma$-plane for analytic initial data. Since the linearization of (a1) is elliptic in $\Gamma$ and $t$ (cf. also [[Elliptic partial differential equation|Elliptic partial differential equation]]), it is hyperbolic in the imaginary $\Gamma$ direction (cf. also [[Hyperbolic partial differential equation|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 [[#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]]].
 
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]]].
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====References====
 
====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>
+
<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>

Latest revision as of 17:00, 1 July 2020

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.

References

[a1] G. Birkhoff, "Helmholtz and Taylor instability" , Proc. Symp. Appl. Math. , XII , Amer. Math. Soc. (1962) pp. 55–76
[a2] R.E. Caflisch, O.F. Orellana, "Long time existence for a slightly perturbed vortex sheet" Commun. Pure Appl. Math. , 39 (1986) pp. 807–838
[a3] R.E. Caflisch, O.F. Orellana, "Singularity formulation and ill-posedness for vortex sheets" SIAM J. Math. Anal. , 20 (1989) pp. 293–307
[a4] J. Duchon, R. Robert, "Global vortex sheet solutions of Euler equations in the plane" J. Diff. Eqs. , 73 (1988) pp. 215–224
[a5] R. Krasny, "On singularity formation in a vortex sheet and the point vortex approximation" J. Fluid Mech. , 167 (1986) pp. 65–93
[a6] R. Krasny, "Desingularization of periodic vortex sheet roll-up" J. Comput. Phys. , 65 (1986) pp. 292–313
[a7] 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
[a8] 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
[a9] 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
[a10] D.I. Pullin, W.R.C. Phillips, "On a generalization of Kaden's problem" J. Fluid Mech. , 104 (1981) pp. 45–53
[a11] N. Rott, "Diffraction of a weak shock with vortex generation" JFM , 1 (1956) pp. 111
[a12] 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
[a13] 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
[a14] 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
[a15] 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
[a16] 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
[a17] M. Brady, A. Leonard, D.I. Pullin, "Regularized vortex sheet evolution in three dimensions" J. Comput. Phys. , 146 (1998) pp. 520–45
How to Cite This Entry:
Birkhoff-Rott equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birkhoff-Rott_equation&oldid=22125
This article was adapted from an original article by Russel E. Caflisch (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article