Some dynamical properties of Ginzburg-Landau vortices. (English) Zbl 0853.35058

We consider the vortex motion for the Ginzburg-Landau heat flow \[ u_t= \Delta u+ {1\over \varepsilon^2} (1- |u|^2)u\quad \text{in } \Omega\times \mathbb{R}_+,\tag{1} \]
\[ (2)\quad u(x, t)= g(x)\quad \text{for } x\in \partial\Omega,\;t> 0,\qquad (3) \quad u(x, 0)= u_0(x)\quad \text{for }x\in \Omega. \] Here \(\Omega\) is a two-dimensional, smooth, bounded domain, \(\varepsilon\) is a positive parameter, \(u: \Omega\times \mathbb{R}_+\to \mathbb{R}^2\), \(g: \partial\Omega\to \mathbb{R}^2\) is smooth, and \(|g|(x)= 1\), \(x\in \partial\Omega\). Naturally, we also assume the compatibility condition that \(u_0(x)= g(x)\) on \(\partial\Omega\). The aim of this article is to understand the global (in time) dynamics of vortices, or zeros, of solutions \(u\) of (1)–(3).
The main results of this paper can be roughly described as follows. Let \(u_\varepsilon(x, t)\) be the solution of (1)–(3), and suppose the initial data satisfy some natural conditions, then, in any finite time \(T\), we show the vortices of \(u_\varepsilon(\cdot, t)\), \(0\leq t\leq T\), remain roughly the same as the initial data, and the phase function of \(u_\varepsilon(\cdot, t)\) satisfies the standard heat equation whenever \(\varepsilon\) is sufficiently small.


35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
80A20 Heat and mass transfer, heat flow (MSC2010)
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