Analysis and Approximation of Gradient Flows Associated with a Fractional Order Gross–Pitaevskii Free Energy

Abstract

We establish the well-posedness of the fractional PDE which arises by considering the gradient flow associated with a fractional Gross–Pitaevskii free energy functional and some basic properties of the solution. The equation reduces to the Allen–Cahn or Cahn–Hilliard equations in the case where the mass tends to zero and an integer order derivative is used in the energy. We study how the presence of a non-zero mass affects the nature of the solutions compared with the Cahn–Hilliard case. In particular, we show that, analogous to the Cahn–Hilliard case, the solutions consist of regions in which the solution is a piecewise constant (whose value depends on the mass and the fractional order) separated by an interface whose width is independent of the mass and the fractional derivative. However, if the average value of the initial data exceeds some threshold (which we determine explicitly), then the solution will tend to a single constant steady state.