Nonlinear Dispersive Equations

2002年3月6日：古斯塔沃·庞塞。DMS-0140023提案标题：非线性色散方程。近年来，对非线性色散方程组解的定性行为的研究取得了重大进展。色散效应和非线性效应之间的关系激发了几项杰出的工作。其中，人们发现尖锐的局部和整体的适定性理论下的最小正则性假设的数据，精确的爆破结果以及不适定性。因此，我们对自然界中的一些现象的理解现在更加完整了。主要研究人员将集中在几个代表性的问题与偏微分方程中产生的波传播和流体力学。波在不同介质中的传播以及不同类型波之间的相互作用可以用复杂的偏微分方程组来模拟。这些系统涉及到许多变量，这些变量取决于问题的特定物理环境。在许多情况下，这些系统过于复杂，无法以合理的精度进行处理，因此必须考虑近似模型。这些近似的正确性对于进一步研究这一物理现象是必不可少的。人们必须给出精确的条件，在这些条件下，这些近似模型的解的定性行为捕获了物理设置的特征。正是在这里，这些解决方案的理论研究变得至关重要。因此，一开始与特殊的解决方案和它的属性，稳定性，长时间的行为，不适定性等情况下，这些理论结果是不可用的一个需要开始研究与适当的数值模拟。

March 6, 2002PI: Gustavo Ponce. DMS-0140023Proposal title : Nonlinear Dispersive Equations. In recent years there has been a significant progress onthe study of qualitative behavior of solution to nonlineardispersive systems. The relation between dispersive and nonlinear effectshas motivated several remarkable works. Among them one findssharp local and global wellposedness theories under minimal regularity assumptions on the data, precise blow up results as well as ill-posedness ones. As a consequence our understanding of several phenomena in nature is now more complete. The principal investigator will focus on severalrepresentative problems connected with partial differentialequations arising in wave propagation and fluid mechanics. The wave propagation in diverse mediums and the interaction ofdifferent kind of waves can be modeled by complicated systems ofpartial differential equations. These systems involve many variableswhich depends of the particular physical setting of the problem. In many cases, these systems are too complicated to be treated with reasonable accuracy, thus one has to consider approximated models. The validity of these approximations is essential to further study of the physical phenomenon. One has to give precise conditions under which the qualitative behavior of the solution of these approximated models captures the features of the physical setting. It is here where the theoretical study of these solutions become essential. Thus, one starts with special solutions and its properties, stability, long time behavior, ill-posedness, etc. In situations where these theoretical results are unavailable one needs to start the study with appropriate numerical simulations.