Refined Jacobian Estimates and Gross–Pitaevsky Vortex Dynamics

Vortex Liquids and the Ginzburg–landau Equation

“…The Neumann boundary condition results are proved in Lemma 10, Lemma 11, and Lemma 13 of [23]; we note a typo in the statement of estimate (38) in [23]. Corresponding results for the Dirichlet boundary condition can be established by using similar arguments.…”

“…We also note from Lemma 14 of [23] that one can easily construct maps u ε (x; a) that satisfy the well-preparedness assumptions (11)- (12). Finally, in the case of a bounded number of vortices, it is well known that the well-preparedness hypothesis is not very important, since one can show that data will become well prepared almost instantaneously due to strong convergence estimates, see [29,21,4,46], and we have no reason to expect a different behavior here.…”

“…Our approach of using differential identities and explicit estimates follows the program of the second author and Jerrard [23] for the Gross-Pitaevsky equation i∂ t u = ∆u + 1 ε 2 u 1 − |u| 2 . Surprisingly, implementing this approach for (1) is more challenging and requires several new estimates.…”

“…In Section 4, we discuss localization estimates for the Jacobian and the energy density. For the Jacobian, results of [23] yield points ξ j such that J (u) − π δ ξ j Ẇ −1,1 is small in a precisely quantified way. We provide a new estimate on the localization of the Ginzburg-Landau energy density to the same set of delta functions of the type e ε (u)…”

“…Using a Hodge decomposition of j (u) |u| − j (u ) and the fact that j (u ) is divergence free while curl( j (u)− j (u ) is controlled, we can bound time averages of terms of the type Ω ρ ζ j (u) |u| − j (u ) for some prescribed function ζ . As in [23], this part is fairly technical, but the differences from the Gross-Pitaevsky case are significant enough that we feel it is necessary to include these details.…”