Vortex dynamics in the presence of excess energy for the Landau–Lifshitz–Gilbert equation

We study the Landau–Lifshitz–Gilbert equation for the dynamics of a magnetic vortex system. We present a PDE-based method for proving vortex dynamics that does not rely on strong well-preparedness of the initial data and allows for instantaneous changes in the strength of the gyrovector force due to bubbling events. The main tools are estimates of the Hodge decomposition of the supercurrent and an analysis of the defect measure of weak convergence of the stress energy tensor. Ginzburg–Landau equations with mixed dynamics in the presence of excess energy are also discussed.

[1]  Fanghua Lin,et al.  Some Dynamical Properties of Ginzburg-Landau Vortices , 1996 .

[2]  S. Serfaty,et al.  Gamma‐convergence of gradient flows with applications to Ginzburg‐Landau , 2004 .

[3]  Y. Meyer,et al.  Compensated compactness and Hardy spaces , 1993 .

[4]  Boling Guo,et al.  The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps , 1993 .

[5]  F. Béthuel,et al.  Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics , 2005 .

[6]  K. Guslienko,et al.  Dynamic origin of vortex core switching in soft magnetic nanodots. , 2007, Physical review letters.

[7]  Peter M. Topping,et al.  Reverse Bubbling and Nonuniqueness in the Harmonic Map Flow , 2002 .

[8]  P. Harpes,et al.  Uniqueness and bubbling of the 2-dimensional Landau-Lifshitz flow , 2004 .

[9]  Halil Mete Soner,et al.  The Jacobian and the Ginzburg-Landau energy , 2002 .

[10]  朴 俊植,et al.  On harmonic maps , 1989 .

[11]  R. Hertel,et al.  Flipping magnetic vortex cores on the picosecond time scale , 2008 .

[12]  Fanghua Lin,et al.  A quantization property for static Ginzburg-Landau vortices , 2001 .

[13]  Sylvia Serfaty,et al.  A product-estimate for Ginzburg–Landau and corollaries , 2004 .

[14]  Gang Tian,et al.  Energy identity for a class of approximate harmonic maps from surfaces , 1995 .

[15]  J. Xin,et al.  On the Incompressible Fluid Limit and the Vortex Motion Law of the Nonlinear Schrödinger Equation , 1999 .

[16]  Fanghua Lin,et al.  Static Theory for Planar Ferromagnets and Antiferromagnets , 2001 .

[17]  S. Serfaty.  Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow Part II: The dynamics , 2007 .

[18]  E Weinan,et al.  Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity , 1994 .

[19]  Chang Kungching,et al.  Heat flow and boundary value problem for harmonic maps , 1989 .

[20]  D. L. Huber,et al.  Dynamics of spin vortices in two-dimensional planar magnets , 1982 .

[21]  J. Eells,et al.  A Report on Harmonic Maps , 1978 .

[22]  Matthias Kurzke,et al.  Vortex Motion for the Landau-Lifshitz-Gilbert Equation with Spin-Transfer Torque , 2011, SIAM J. Math. Anal..

[23]  Matthias Kurzke,et al.  Ginzburg–Landau Vortices Driven by the Landau–Lifshitz–Gilbert Equation , 2011 .

[24]  Matthias Kurzke,et al.  Dynamics for ginzburg-landau vortices under a mixed flow , 2009 .

[25]  H. Brezis,et al.  Ginzburg-Landau Vortices , 1994 .

[26]  Halil Mete Soner,et al.  Dynamics of Ginzburg‐Landau Vortices , 1998 .

[27]  A. Freire.  Uniqueness for the harmonic map flow from surfaces to general targets , 1995 .

[28]  Damped wave dynamics for a complex Ginzburg-Landau equation with low dissipation , 2010, 1003.5375.

[29]  S. Serfaty.  Vortex collisions and energy-dissipation rates in the Ginzburg–Landau heat flow. Part I: Study of the perturbed Ginzburg–Landau equation , 2007 .

[30]  Ling Fang-hua,et al.  Soliton dynamics in planar ferromagnets and anti-ferromagnets , 2003 .

[31]  Michiel Bertsch,et al.  Nonuniqueness for the Heat Flow¶of Harmonic Maps on the Disk , 2002 .

[32]  F. Lin,et al.  Traveling wave solutions of the Schrödinger map equation , 2010 .

[33]  Robert L. Jerrard,et al.  Ginzburg-landau vortices: weak stability and schrödinger equation dynamics , 1999 .

[34]  Michael Struwe,et al.  On the evolution of harmonic mappings of Riemannian surfaces , 1985 .

[35]  Frédéric Hélein,et al.  Harmonic Maps, Conservation Laws, And Moving Frames , 2002 .

[36]  Melanie Rupflin.  An improved uniqueness result for the harmonic map flow in two dimensions , 2008 .

[37]  A. Thiele.  Steady-State Motion of Magnetic Domains , 1973 .

[38]  Evelyne Miot,et al.  Dynamics of vortices for the complex Ginzburg–Landau equation , 2008, 0810.4782.