Motion of a vortex filament in an external flow

We consider a nonlinear model equation describing the motion of a vortex filament immersed in an incompressible and inviscid fluid. In the present problem setting, we also take into account the effect of external flow. We prove the unique solvability, locally in time, of an initial value problem posed on the one dimensional torus. The problem describes the motion of a closed vortex filament.

[1]  N. Tsuge.  Isentropic Gas Flow for the Compressible Euler Equation in a Nozzle , 2013 .

[2]  N. Tsuge.  Existence of Global Solutions for Unsteady Isentropic Gas Flow in a Laval Nozzle , 2012 .

[3]  T. Iguchi,et al.  Motion of a Vortex Filament in the Half Space , 2010, 1006.3905.

[4]  E. Onodera.  A Remark on the Global Existence of a Third Order Dispersive Flow into Locally Hermitian Symmetric Spaces , 2009, 0906.3171.

[5]  Luis Vega,et al.  Scattering for 1D cubic NLS and singular vortex dynamics , 2009, 0905.0062.

[6]  Bing-Yu Zhang,et al.  Non-homogeneous boundary value problems for the Korteweg–de Vries and the Korteweg–de Vries–Burgers equations in a quarter plane , 2008 .

[7]  J. Segata.  On asymptotic behavior of solutions to Korteweg–de Vries type equations related to vortex filament with axial flow , 2008 .

[8]  Luis Vega,et al.  On the Stability of a Singular Vortex Dynamics , 2008, 0802.1996.

[9]  E. Onodera.  A Third-Order Dispersive Flow for Closed Curves into Kähler Manifolds , 2007, 0707.2660.

[10]  N. Hayashi,et al.  Neumann problem for the Korteweg–de Vries equation , 2006 .

[11]  L. Berselli,et al.  On the Global Evolution of Vortex Filaments, Blobs, and Small Loops in 3D Ideal Flows , 2005, math-ph/0510091.

[12]  Luis Vega,et al.  Formation of Singularities and Self-Similar Vortex Motion Under the Localized Induction Approximation , 2003 .

[13]  L. Berselli,et al.  Some results for the line vortex equation , 2002 .

[14]  H. R. Ruiz Paredes,et al.  Boundary-value problem for the Korteweg-de Vries-Burgers type equation , 2001 .

[15]  N. Koiso.  The vortex filament equation and a semilinear Schrödinger equation in a Hermitian symmetric space , 1997 .

[16]  A. Tani,et al.  Initial and Initial-Boundary Value Problems for a Vortex Filament with or without Axial Flow , 1996 .

[17]  A. Tani,et al.  Solvability of the localized induction equation for vortex motion , 1994 .

[18]  Tai-Ping Liu.  Quasilinear hyperbolic systems , 1979 .

[19]  F. Massey,et al.  Differentiability of solutions to hyperbolic initial-boundary value problems , 1974 .

[20]  H. Hasimoto,et al.  A soliton on a vortex filament , 1972, Journal of Fluid Mechanics.

[21]  F. R. Hama,et al.  Localized‐Induction Concept on a Curved Vortex and Motion of an Elliptic Vortex Ring , 1965 .

[22]  Louis Rosenhead,et al.  The Spread of Vorticity in the Wake Behind a Cylinder , 1930 .

[23]  Luigi Sante Da Rios.  Sul moto d’un liquido indefinito con un filetto vorticoso di forma qualunque , 1906 .

[24]  T. Nishiyama.  On the motion of a vortex filament in an external flow according to the localized induction approximation , 1999, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.