Vortices in the Magnetic Ginzburg-Landau Model pp 165-206 | Cite as

# The Intermediate Regime

## Abstract

When *h*_{ex} ∼ *H* _{c1} ^{0} i.e., \(
\frac{{h_{ex} }}
{{\left| {log \varepsilon } \right|}} \to \lambda = \frac{1}
{{2\left| {\underline {\xi _0 } } \right|}}
\)
, then from Theorem 7.2, we get that the limiting minimizer is *h*_{0} hence μ* = 0. Moreover, comparing the lower bounds (7.58) and (7.59) to the upper bound of Theorem 7.1, we find \(
\frac{{\sum\nolimits_i {\left| {d_i } \right|} }}
{{h_{ex} }} \to 0
\)
, which means that the number of vortices is *o*(*h*_{ex}). In other words, for energy-minimizers, vortices first appear for \(
\frac{{h_{ex} }}
{{\left| {log \varepsilon } \right|}} \to \frac{1}
{{2\left| {\underline {\xi _0 } } \right|}}
\)
, or *h*_{ex} ∼ *H* _{c1} ^{0} , and next to Λ (defined in (7.5)), and the vorticity mass is much smaller than *h*_{ex}. The analysis of Chapter 7 does not give us the optimal number *n* of vortices nor the full asymptotic expansion of the first critical field. Thus, a more detailed study will be necessary in this regime \(
h_{ex} \sim \frac{{\left| {\left. {log \varepsilon } \right|} \right.}}
{{2\left| {\left. {\xi _0 } \right|} \right.}}
\)
, in which *n* ≪ *h*_{ex}. We will prove that the vortices, even though their number may be diverging, all concentrate around Λ (generically a single point) but that after a suitable blow-up, they tend to arrange in a uniform density on a subdomain of ℝ^{2}, in order to minimize a limiting interaction energy *I* defined on probability measures.

## Keywords

Probability Measure Small Ball Landau Equation Intermediate Regime Large Ball## Preview

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