Mathematical Sciences: Nonlinear Functional Analysis in Applied Science

小行星9401418 该项目涉及现代泛函分析方法的发展和应用，以解决应用科学中一些普遍接受的数学模型中出现的问题。研究者建议继续发展现代泛函分析技术，这些技术适合于研究单调动力系统解的大时间渐近行为。一个特别强调这一部分的拟议研究将在大时间渐近行为的轨道（轨迹）在几乎周期单调动力学过程。各种不稳定现象描述的抛物型系统的复杂的金兹伯格-朗道方程。研究者提出建立抛物系统解的局部（在时间上）存在性、唯一性和复解析延拓。随后将在多维复空间中的区域上的Lebesgue-Hardy空间中发展合适的全局估计，以防止解的所有导数在有限时间内爆破。 对于金兹伯格-朗道方程，应该可以直接从方程中得到这样的估计。 数学模型解的全局存在性、唯一性和复解析延拓保证了该模型所描述的特定物理（生物、物质或经济）结构在所有时间的存在性和唯一性，以及它对空间和时间变量的平滑依赖性（没有突然的跳跃）。这些特性对于在物理结构上执行的测量的设计是必不可少的。模型解的大时间渐近行为决定了模型结构内关系的最终状态。 本计画所研究之数学模型均以非线性微分方程式表示。 ***

9401418 Takac This project is concerned with development and applications of methods of modern functional analysis to problems arising in a number of generally well-accepted mathematical models in applied science. The investigator proposes to continue his development of modern functional analytic techniques that are suitable for studying large-time asymptotic behavior of solutions to monotone dynamical systems. A particular emphasis of this part of the proposed research will be on the large-time asymptotic behavior of orbits (trajectories) in almost periodic monotone dynamical processes. Various instability phenomena are described by a parabolic system of complex Ginzburg-Landau equations. The investigator proposes to establish the local (in time) existence, uniqueness and complex analytic extension of solutions to the parabolic system. This will be followed by a development of suitable global estimates in the Lebesgue-Hardy spaces over a region in multi-dimensional complex space in order to prevent all the derivatives of the solution from blowing-up in finite time. For the Ginzburg-Landau equation it should be possible to obtain such estimates directly from the equation. Global existence, uniqueness and complex analytic extension of solutions to a mathematical model guarantee the existence and uniqueness for all times of a particular physical (biological, material or economic) structure described by this model, and its smooth dependence upon the spatial and temporal variables (no sudden jumps). These properties are essential for the design of measurements to be performed on the physical structure. Large- time asymptotic behavior of solutions to a model determines the ultimate state of relationships within the modeled structure. Mathematical models investigated in this project are expressed in terms of non-linear differential equations. ***