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

作     者：Mark Ainsworth Zhiping Mao 

作者机构：Division of Applied MathematicsBrown UniversityProvidenceRI 02912USA 

出 版 物：《Communications on Applied Mathematics and Computation》 (应用数学与计算数学学报（英文）)

年 卷 期：2019年第1卷第1期

页      面：5-19页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：This work was supported by the MURI/ARO on "Fractional PDEs for Conservation Laws and Beyond: Theory  Numerics and Applications" (W911NF-15-1-0562) 

主　　题：Fractional differential equation Non-local energy Well-posedness Fourier spectral method 

摘      要：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 *** 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 *** study how the presence of a non-zero mass affects the nature of the solutions compared with the Cahn-Hilliard *** 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 ***,if the average value of the initial data exceeds some threshold(which we determine explic让ly),then the solution will tend to a single constant steady state.