## 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.

### MSC:

 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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