Nonlinear Evolution Equations

拟议的研究将侧重于涉及流体力学和波传播研究中出现的非线性偏微分方程系统的数学模型。工作的重点将是与任何这样的系统的初始值问题的解决方案的定性性质。利用色散关系和非线性结构之间的关系，将研究非线性色散模型的独特连续性。一个具体的问题将是建立一个解决方案的唯一性，以及它的可能重建从一些不完整的信息在两个不同的时间。关于几种涡度形式的时间演变的其他问题也将被考虑。作为经典决定论原理的结果，物理系统的演化规律与初始配置一起应该描绘系统在所有未来时间的状态。用数学术语来说，物理定律被编码在一个偏微分方程系统中，该系统的解应该存在，并且唯一且连续地依赖于初始配置。因此，一个数学框架的发展，其中的存在性，唯一性和稳定性的解决方案，以一个给定的系统可以验证和量化是一个问题的根本重要性。数学分析提供了一个强有力的工具来研究这些模型的解的性质，远远超出了纯现象学假设的范围。这个项目的大部分研究都致力于描述水波和不可压缩流体动力学模型的基本问题。

The proposed research will focus on mathematical models involving systems of nonlinear partial differential equations that arise in fluid mechanics and in the study of wave propagation. The emphasis of the work will be on the qualitative properties of solutions to the initial value problem associated with any such system. Unique continuation properties of nonlinear dispersive models will be investigated by exploiting the relationship between the dispersive relation and the structure of the nonlinearity. A specific problem will be to establish the uniqueness of a solution, as well as its possible reconstruction from some incomplete information given at two different times. Other problems concerning the time evolution of several forms of the vorticity will be also considered. As a consequence of the classical principle of determinism, the evolutionary laws of a physical system together with an initial configuration should delineate the state of the system at all future times. To rephrase this in mathematical terms, the laws of physics are encoded in a system of partial differential equations whose solutions should exist and depend uniquely and continuously on the initial configuration. Hence, the development of a mathematical framework in which the existence, uniqueness, and stability of the solution to a given system can be verified and quantified is a problem of fundamental importance. Mathematical analysis offers a powerful tool for investigating the properties of solutions to these models well beyond the point of purely phenomenological assumptions. Much of the research in this project is devoted to such basic questions for models describing the dynamics of water waves and incompressible fluids.