Mathematical Sciences: Operator Theory and Differential Equations

本项目继续将一般算子理论泛函分析与微分方程相结合的数学研究。在工作中将处理三个重叠的区域。第一类是空间变量的混合型非线性演化偏微分方程。最初的工作将集中在一个具体的例子上，即等离子体物理学的Kompaneetz方程。它可以写成一个抽象的柯西问题。第一个任务是确定溶剂的性质；特别是它是否有一个密集的范围。第二个领域涉及算子的半群，特别是那些其无穷小发生器分裂成线性和局部光滑非线性部分的算子。目的是确定一个全局定义的半群算子在任何时间都能确定解的条件，并将任意两个解的差作为时间和初始函数范数的函数进行定界。最后，将研究自旋极化托马斯-费米理论，涉及函数对的最小化问题。寻找极值的一个主要障碍是涉及两个未知数的泛函不是凸的。通过拓扑论证，可以证明极值是存在的。下一步将是找到一种方法，以具体的方式产生最小值。该项目将抽象的数学概念与物理世界中各种现象的模型相结合。研究的目标之一是将模型拟合到适当的广义结构中，并应用现有的理论。通过这种方式，人们可以深入了解模型，有时还可以发现隐藏的属性。

This project continues mathematical research combining general operator-theoretic functional analysis with differential equations. Three overlapping areas will be treated in the work. The first concerns nonlinear partial differential equations of evolution which are of mixed type in the space variable. Initial work will focus on a concrete example, the Kompaneetz Equation of plasma physics. It can be written as an abstract Cauchy problem. The first task is to determine properties of the resolvent; in particular whether or not it has a dense range. The second area involves semigroups of operators, particularly those whose infinitesimal generators split into a linear and locally smooth nonlinear part. The object is to determine conditions for a globally defined semigroup of operators determining the solution for all time and to bound the difference of any two solutions as a function of time and the initial function norms. Finally, work will be done on spin polarization Thomas-Fermi theory involving a minimizing problem for pairs of functions. One main obstacle to finding extrema is the functional involving the two unknowns is not convex. By topological arguments, one can show that extrema exist. The next step will be to find a method which will produce the minima in a concrete fashion. The project combines abstract mathematical concepts with models of various phenomena in the physical world. Among the goals of the research is one of fitting models into the proper generalized structures and apply the existing theory. In this way one gains insight into the models and, at times, discovers hidden properties.