Nonlinear Evolution Equations

主要的研究人员将集中在与波动传播和流体力学中的偏微分方程组有关的几个具有代表性的问题上。近年来，对非线性色散模型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一些杰出的工作。因此，我们现在对自然界中的几个现象的理解更加完整。然而，有几个基本问题仍然悬而未决。例如，数学和物理中出现的几个薛定谔类型的系统的柯西问题的存在理论(局部和全局)是不可用的。这是由于经典方法不能应用的事实。因此，人们不得不发展其他方法，这些方法在很大程度上依赖于谐波分析的技术和思想。与C.E.Kenig和L.Vega合作，首席研究员计划继续发展傅立叶分析这一非线性偏微分方程组的基本领域。波在不同介质中的传播和不同类型波的相互作用可以用复杂的偏微分方程组来模拟。这些系统涉及许多变量，这取决于问题的特定物理环境。在许多情况下，这些系统太复杂，不能以合理的精度处理，因此必须考虑近似模型。这些近似的有效性对于进一步研究物理现象是至关重要的。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捉到了物理环境的特征。正是在这里，对这些解决方案的理论研究变得至关重要。因此，人们从特解及其性质、稳定性、长时间行为、不适定性等开始。在这些理论结果不可用的情况下，需要从一些数值模拟开始研究

The principal investigator will focus on severalrepresentative problems connected with partial differentialequations arising in wave propagation and fluid mechanics.In recent years there has been a significant progress on the study of qualitative behavior of solution to nonlineardispersive models. The relation between dispersive and nonlinear effects has motivated several remarkable works. As a consequence our understandingof several phenomena in nature is now more complete.However, several essential question remain open. For example, an existence theory (local and global) for the Cauchy problem for several systems of Schr\"odinger type arising in both mathematics and physics is unavailable. This is due to the fact that classical methods can not be applied. Thus one has to develop other approachs which rely heavily on techniques and ideas in harmonic analysis. In collaboration with C. E. Kenig and L. Vega, the principal investigator plans to continue the development of Fourier Analysis for this fundamental area of nonlinear PDE's. The wave propagation in diverse mediums and the interaction of different kind of waves can be modeled by complicated systems of partial 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 vality 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 ofthese 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 some numerical simulations