Mathematical Foundation

Abstract

In this chapter, we discuss a simplified Landau phase transition model by setting the magnetic potential in the superconductivity problem to be zero. The aim is to concentrate on the mathematical issues involved in describing the phase transition phenomena associated with the model. The G-L energy we look at takes the form

$$E(u) = \frac{1}{2}\int_\Omega {|\nabla u{|^2} + \frac{1}{2}{{(|u{|^2} - 1)}^2}dx.}$$

(0.1)

The associated steady state PDE is

$$- \Delta u + (|u{|^2} - 1)u = 0$$

(0.2)

and the associated evolutionary PDE is

$${u_t} - \Delta u + (|u{|^2} - 1)u = 0$$

(0.3)

Because the solutions to this equation change their value slowly, to see rapid phase changes around the points where the solutions take up the value 0, the energy is supposed to take the form

$${E_\varepsilon }(u) = \frac{1}{2}\int_\Omega {|\nabla u{|^2} + \frac{1}{{2{\varepsilon ^2}}}{{(|u{|^2} - 1)}^2}dx}$$

with ε a small parameter. Here Ω is a bounded, smooth domain in ℝⁿ with n = 2 or 3. The phase transition phenomenon appears over a length scale O(ε^1/2) because when u takes up the value zero, the small ɛ will force it rapidly back to the unit circle. This ε dependent equation can be obtained from the original equation via a scaling of the form (disregard the change of the underlying domain)

$$ t' = {\varepsilon ^2}t, x' = \varepsilon x. $$

The fact that the resealed model is effective in deriving the asymptotic equations of phase changes suggests that the original Landau model describes slow moving and slow varying phase transition phenomenon.