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

- 1 Downloads

## 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.

## Keywords

Fractional differential equation Non-local energy Well-posedness Fourier spectral method## Mathematics Subject Classification

65N12 65N30 65N50## References

- 1.Ainsworth, M., Mao, Z.: Analysis and approximation of a fractional Cahn–Hilliard equation. SIAM J. Numer. Anal.
**55**, 1689–1718 (2017)MathSciNetCrossRefGoogle Scholar - 2.Ainsworth, M., Mao, Z.: Well-posedness of the Cahn–Hilliard equation with fractional free energy and its Fourier Galerkin approximation. Chaos Solitons Fractals.
**102**, 264–273 (2017)MathSciNetCrossRefGoogle Scholar - 3.Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall.
**27**, 1085–1095 (1979)CrossRefGoogle Scholar - 4.Ambrosio, V.: On the existence of periodic solutions for a fractional Schrödinger equation. In: Proceedings of the American Mathematical Society (2018)Google Scholar
- 5.Antoine, X., Tang, Q., Zhang, Y.: On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross–Pitaevskii equations with rotation term and nonlocal nonlinear interactions. J. Comput. Phys.
**325**, 74–97 (2016)MathSciNetCrossRefGoogle Scholar - 6.Bogdan, K., Byczkowski, T., Kulczycki, T., Ryznar, M., Song, R., Vondracek, Z.: Potential analysis of stable processes and its extensions. In: Graczyk, P., Stos, A. (eds.) Lecture Notes in Mathematics, vol. 1980. Springer, Berlin (2009)Google Scholar
- 7.Cahn, J.W.: Free energy of a nonuniform system. II. Thermodynamic basis. J. Chem. Phys.
**30**, 1121–1124 (1959)CrossRefGoogle Scholar - 8.Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys.
**28**, 258–267 (1958)CrossRefGoogle Scholar - 9.Kopriva, D.A.: Implementing spectral methods for partial differential equations: algorithms for scientists and engineers. Springer, Berlin (2009)CrossRefGoogle Scholar
- 10.Lieb, E.H., Yau, H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Comm. Math. Phys.
**112**, 147–174 (1987)MathSciNetCrossRefGoogle Scholar - 11.Lieb, E.H., Yau, H.-T.: The stability and instability of relativistic matter. Comm. Math. Phys.
**118**, 177–213 (1988)MathSciNetCrossRefGoogle Scholar - 12.Shen, J., Yang, X.: Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discrete Cont. Dyn. Syst.
**28**, 1669–1691 (2010)MathSciNetCrossRefGoogle Scholar - 13.Yue, P., Feng, J.J., Liu, C., Shen, J.: A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech.
**515**, 293–317 (2004)MathSciNetCrossRefGoogle Scholar