We first provide an overview of gradient flow for minimization problem.  Then we note that the ground state for the ultracold fermi gas with dipole-dipole interaction is a functional minimization problem based on density functional theory (DFT). We extend the recent work on Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem, and present continuous projected Sobolev gradient flows for computing the DFT-based ground state solution of ultracold dipolar fermi gas. We prove that the gradient flows have the properties of orthonormal preserving and energy diminishing, which is desirable for the computation of the ground state solution. Many numerical technique for partial differential equation can be used to discretize the time-dependant projected Sobolev gradient flows, which may be an advantage of the method. We propose an efficient and accurate numerical scheme – semi-implicit Euler method in time and Fourier spectral method in space for discretizing these projected Sobolev gradient flows and use them to find the ground states of the fermi gas numerically. Extensive numerical examples in three dimensions for ground states are reported to demonstrate the power of the numerical methods and to discuss the physics of dipolar fermi gas at very low temperature.

王汉权，云南财经大学统计与数学学院教授，博士生导师，中国数学会计算数学分会常务理事，云南省中青年学术带头人，云南省数学会副理事长，云南省工业与应用数学会副理事长，四川大学数学学院博士生导师。2013年入选教育部“新世纪优秀人才支持计划”，2014年获“云南省有突出贡献优秀专业技术人才”称号，2017年获云南省自然科学奖三等奖。主持国家自然科学基金 5项。在国内外学术期刊上发表高水平研究论文60余篇，出版专著2部、教材2本。