Mathematical Sciences: Nonlinear Evolutions Equations III

该奖项支持的研究将集中在各种 非线性发展方程：非线性部分 解随时间变化的微分方程。 工作 将经典微分方程理论与调和 分析和数学物理。 主要研究领域 包括近似产生非线性色散系统 非线性波的传播模型。 特别感兴趣 将致力于确定Sobolev指数的下界 这保证了局部适定性， 广义Korteweg-DeVries方程 类似的努力将是 应用于非线性薛定谔方程的研究。 无论是 案件的初步工作已经改善了已知的估计， 指数。还将分析高阶方程。 这里，方程的非线性部分将被限制为 没有常数项的多项式。 这类进化 方程模拟弹性介质中的波。当地的结果有 已经建立了平滑的解决方案，但 长期行为还不太清楚。 例如，第五 阶系统没有孤波解。额外 计划在Zakharov-Schulman系统上进行工作，该系统描述了 小振幅高频波与 声波类型当线性部分是椭圆形时， 方程已被广泛研究，否则没有结果， 知道的 偏微分方程的基础是 物理世界的数学模型。 的作用 数学分析并不是为了创造方程， 它是提供关于以下方面的定性和定量信息： 解决方案。 这可能包括以下问题的答案 独特性、平滑性和成长性。 此外，分析往往 发展的解决方案和估计的近似方法 这些近似值的准确性。

Research supported by this award will focus on various aspects of nonlinear evolution equations: nonlinear partial differential equations whose solutions evolve in time. The work combines classical theory of differential equations with harmonic analysis and mathematical physics. The principal areas of study include nonlinear dispersive systems arising as approximate models in propagation of nonlinear waves. Particular interest will be paid to determining the lower bound of Sobolev exponents which guarantee local well-posedness for the generalized Korteweg-deVries equation. A similar effort will be applied to the study of nonlinear Schrodinger equations. In both cases preliminary work has already improved known estimates on the exponents. Higher order equations will also be analyzed. Here the nonlinear part of the equations will be restricted to polynomials having no constant term. This class of evolution equation models waves in elastic media. Local results have already been established for the smoothness of solutions, but long-time behavior is not well understood. For instance, fifth order systems do not have solitary wave solutions. Additional work is planned on the Zakharov-Schulman systems which describe interactions of small-amplitude, high frequencey wave with acoustic type waves. Where the linear part is elliptic, the equations have been extensively studied, otherwise no results are known. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.