Mathematical Sciences: Non-Linear Evolution Equations

8902704 庞塞 本报告将强调三个主要主题 数学项目，组合部分 微分方程、调和分析和数学 物理学 第一个是非线性演化方程 专注于涉及分数的交换子估计 衍生物. 这些估计用于获得 双线性拟微分的局部到全局信息 运算符，例如， Navier-Stokes方程可以无限延拓，如果 变形张量和涡度保持有限。 还将开展工作， 解的局部正则性和渐近性 非线性色散方程 具体实例包括 广义Korteweg-de弗里斯方程，其中没有 已知的全局存在唯一性结果。而且，数值 计算表明，当幂 方程中的函数超过5。 第三个领域涉及柯西问题， 三维热弹性系统（对于均匀 中等）。方程组形成了一个耦合的双曲-抛物方程组。 系统 这项研究的一个直接目标是确定 小溶液的行为是否受 双曲线部分或抛物线部分。 对于纯弹性 在这种情况下，这个问题已经研究过了，但是， 所使用的技术不适用于本发明。 以后的工作将考虑的问题的特点， 可能导致奇异性的系统非线性项 或具有小初始数据的解决方案。

8902704 Ponce Three primary themes will be emphasized in this mathematical project, combining elements of partial differential equations, harmonic analysis and mathematical physics. The first involves nonlinear evolution equations focusing on commutator estimates involving fractional derivatives. These estimates are used to obtain local-to-global information on bilinear pseudodifferential operators illustrating, for example, how Euler and Navier-Stokes equations can be continued indefinitely if the deformation tensor and vorticity remains finite. Work will also be done in developing information about the local regularity and asymptotic behavior of solutions to nonlinear dispersive equations. Specific examples include the generalized Korteweg-de Vries equations for which there are no known global existence-uniqueness results. Moreover, numerical calculations suggest a blow-up of solutions when the power of the function in the equation exceeds five. The third area concerns the Cauchy problem for the three-dimensional thermoelasticity system (for a homogeneous medium). The equations form a coupled hyperbolic-parabolic system. An immediate goal of the research is to determine whether the behavior of small solutions is controlled by the hyperbolic part or by the parabolic part. For the pure elastic case, this problem has been studied earlier, however the techniques used are not applicable in the present context. Later work will consider the question of characterizing the nonlinear terms of the systems which can lead to singularities or solutions which have small initial data.