Topological Singularities in Periodic Media: Ginzburg–Landau and Core-Radius Approaches

We describe the emergence of topological singularities in periodic media within the Ginzburg–Landau model and the core-radius approach. The energy functionals of both models are denoted by $$E_{\varepsilon ,\delta }$$ E ε , δ , where $$\varepsilon $$ ε represent the coherence length (in the Ginzburg–Landau model) or the core-radius size (in the core-radius approach) and $$\delta $$ δ denotes the periodicity scale. We carry out the $$\Gamma $$ Γ -convergence analysis of $$E_{\varepsilon ,\delta }$$ E ε , δ as $$\varepsilon \rightarrow 0$$ ε → 0 and $$\delta =\delta _\varepsilon \rightarrow 0$$ δ = δ ε → 0 in the $$|\log \varepsilon |$$ | log ε | scaling regime, showing that the $$\Gamma $$ Γ -limit consists in the energy cost of finitely many vortex-like point singularities of integer degree. After introducing the scale parameter $$\begin{aligned} \lambda =\min \Bigl \{1,\lim _{\varepsilon \rightarrow 0} {|\log \delta _\varepsilon |\over |\log \varepsilon |}\Bigr \} \end{aligned}$$ λ = min { 1 , lim ε → 0 | log δ ε | | log ε | } (upon extraction of subsequences), we show that in a sense we always have a separation-of-scale effect: at scales smaller than $$\varepsilon ^\lambda $$ ε λ we first have a concentration process around some vortices whose location is subsequently optimized, while for scales larger than $$\varepsilon ^\lambda $$ ε λ the concentration process takes place “after” homogenization.

ENERGY CONCENTRATION PROPERTIES OF A p-GINZBURG–LANDAU MODEL

This paper is concerned with the p-Ginzburg–Landau (p-GL) type model with $p\neq 2$ . First, we obtain global energy estimates and energy concentration properties by the singularity analysis. Next, we give a decay rate of $1-|u_\varepsilon |$ in the domain away from the singularities when $\varepsilon \to 0$ , where $u_\varepsilon $ is a minimizer of p-GL functional with $p \in (1,2)$ . Finally, we obtain a Liouville theorem for the finite energy solutions of the p-GL equation on $\mathbb {R}^2$ .

A theory of giant vortices

We elaborate a theory of giant vortices [1] based on an asymptotic expansion in inverse powers of their winding number n. The theory is applied to the analysis of vortex solutions in the abelian Higgs (Ginzburg-Landau) model. Specific properties of the giant vortices for charged and neutral scalar fields as well as different integrable limits of the scalar self-coupling are discussed. Asymptotic results and the finite-n corrections to the vortex solutions are derived in analytic form and the convergence region of the expansion is determined.

An energy and momentum conserving collisional bracket for the guiding-centre Vlasov–Maxwell–Landau model

This paper proposes a metric bracket for representing Coulomb collisions in the so-called guiding-centre Vlasov–Maxwell–Landau model. The bracket is manufactured to preserve the same energy and momentum functionals as does the Vlasov–Maxwell part and to simultaneously satisfy a revised version of the H-theorem, where the equilibrium distributions with respect to collisional dynamics are identified as Maxwellians. This is achieved by exploiting the special projective nature of the Landau collision operator and the simple form of the system's momentum functional. A discussion regarding a possible extension of the results to electromagnetic drift-kinetic and gyrokinetic systems is included. We anticipate that energy conservation and entropy dissipation can always be manufactured whereas guaranteeing momentum conservation is a delicate matter yet to be resolved.