Dynamics of multiple degree Ginzburg-Landau vortices. (English) Zbl 1135.35014

Bethuel, F.; Orlandi, G.; Smets, D.

Commun. Math. Phys. 272, No. 1, 229-261 (2007).

This paper deals with a careful asymptotic analysis as \(\varepsilon\rightarrow 0\) of the complex-valued parabolic Ginzburg-Landau equation \(\frac{\partial u_\varepsilon}{\partial t}-\Delta u_\varepsilon =\frac{1}{\varepsilon^2}u_\varepsilon (1-| u_\varepsilon | ^2)\) on \(\mathbb R^2\times \mathbb R^+\). The authors focus on the description of the asymptotic behavior of solutions as \(\varepsilon\rightarrow 0\), under the only assumption that the initial datum \(u^0_\varepsilon (\cdot )\equiv u_\varepsilon (\cdot ,0)\) verifies \(E_\varepsilon (u^0_\varepsilon )\leq M_0| \log\varepsilon | \), where \(E_\varepsilon\) stands for the associated energy functional.
The main result of this paper provides a complete description of the trajectory set, in the sense that, asymptotically, vortices evolve according to a simple ordinary differential equation, which is a gradient flow of the Kirchhoff-Onsager functional, defined as \(W(a_1,\ldots ,a_n)=-2\sum_{i\not=j=1}^nd_id_j\log | a_i-a_j| \). Next, the authors study the asymptotic behavior of the trajectories near a branching point and show that, after a suitable parabolic rescaling centered at the collision point, vortices converge to the set of critical points of \(W\) restricted to a certain prescribed manifold. The proofs are based on powerful asymptotic estimates combined with original analytic arguments.

Reviewer: Vicenţiu D. Rădulescu (Craiova)

MSC:

35B25	Singular perturbations in context of PDEs
58E50	Applications of variational problems in infinite-dimensional spaces to the sciences
35K55	Nonlinear parabolic equations

Keywords:

asymptotic analysis; trajectory set; Kirchhoff-Onsager functional

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