Some Dynamical Properties of Ginzburg-Landau Vortices

Here 0 is a two-dimensional, smooth, bounded domain, E is a positive parameter, u : R x R, R2, g : 8 R R2 is smooth, and Igl(x) = l , x E 80. Naturally we also assume the compatibility condition that u&) = g(x) on dfl. The system (l.lH1.3) can be viewed as a simplified evolutionary GinzburgLandau equation in the theory of superconductivity (141, [51, 1111, [18]). The same system also appears in a canonical way when one expands a large class of second-order dissipative systems about bifurcation points (131, [15], [19]). It serves, therefore, as one of the fundamental models in the study of the dynamics of nonequilibrium patterns ([21], [22]). The aim of this article is to understand the global (in time) dynamics of vortices, or zeros, of solutions u of (1.1)-(1.3). Our study has some interesting implications for the problem of “pinning the Ginzburg-Landau vortices”; see, for example, [6] and [16]. Its importance to the theory of superconductivity and applications are addressed in many earlier works ( [6] , [7], [lo], [141, [181). To understand the behavior of solutions u of ( l . lHl .3) as t +m, one has to look at steady state solutions uE, that is, the critical points of the energy functional