Mathematical Sciences: Variational Methods and Applications

8901477奥赫穆蒂首席调查员将对某些在应用中很重要的问题进行变分原理的构造和分析研究。他将开发和实施一些新的算法来数值求解这些变分问题。所要研究的问题包括非自伴椭圆本征问题及其在线性流体动力稳定性理论中的应用，以及量子理论中出现的Hartree和Hartree-Fock方程的非线性本征值问题。另一类问题集中在Navier-Stokes方程解的变分刻画上，包括这些原理的性质和结果。数值研究将集中在寻找对称和非对称矩阵的特征值和特征向量的数值方法，以及旋转自引力流体平衡的数值模拟，这是恒星和行星的简单流体动力学模型。

8901477 Auchmuty The principal investigator will conduct research on the construction and analysis of variational principles for certain problems that are important in applications. He will develop, and implement, some new algorithms for numerically solving these variational problems. The problems to be studied include non-self-adjoint elliptic eigenproblems and their application to linear hydrodynamic stability theory, and nonlinear eigenvalue problems for the Hartree and Hartree-Fock equations which arise in quantum theory. Another class of problems centers on the variational characterization of solutions of the Navier-Stokes equations, including the properties and consequences of these principles. The numerical research will center on numerical methods for finding eigenvalues and eigenvectors of both symmetric and non-symmetric matrices, also the numerical simulation of the equilibria of rotating self-gravitating fluids, which are simple hydrodynamical models of stars and planets.