Mathematical Sciences: Operator Theory and Differential Equations

这个项目继续数学研究结合 一般算子论泛函微分分析 方程 工作中将处理三个重叠的领域。 第一个是非线性偏微分方程， 在空间变量上表现为混合型演化。 初始 工作将集中在一个具体的例子，Kompaneetz方程的 等离子体物理学 它可以写成一个抽象的柯西问题。 第一个任务是确定预解式的性质; 特别是它是否有密集的范围。 第二区域 涉及算子半群，特别是那些 无穷小生成元分裂成线性和局部光滑 非线性部分 目的是确定 整体定义的算子半群决定解 并将任何两个解的差限制为 时间的函数和初始函数范数。最后，工作 将在自旋极化的费米子理论，涉及一个 函数对的最小化问题 一个主要障碍是 找到极值是涉及两个未知数的泛函， 不是凸的。 通过拓扑论证，我们可以证明， 存在. 下一步将是找到一种方法， 具体的最小值。 该项目将抽象的数学概念与 物理世界中各种现象的模型。 中 研究的目的是将模型拟合到适当的 并应用现有的理论。 在这 一个人深入了解模型的方式，有时， 隐藏属性

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.