When hexH 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 h0 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(hex). 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 hexH c1 0 , and next to Λ (defined in (7.5)), and the vorticity mass is much smaller than hex. 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 nhex. 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 ℝ2, in order to minimize a limiting interaction energy I defined on probability measures.


Probability Measure Small Ball Landau Equation Intermediate Regime Large Ball 
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© Birkhäuser Boston 2007

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