Nonlinear Evolution Equations

主要研究者将集中在波动传播和流体力学中与偏微分方程相关的几个代表性问题上，近年来，非线性色散模型解的定性行为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个显着的工作。结果，我们对自然界的一些现象有了更全面的了解，然而，一些基本问题仍然没有解决。例如，在数学和物理中出现的几个Schr\“odinger型系统的Cauchy问题的存在性理论（局部和整体）是不可用的。这是由于经典方法不能应用的事实。因此，人们必须开发其他方法，这些方法在很大程度上依赖于谐波分析中的技术和思想。与C合作。E. Kenig和L.维加，首席研究员计划继续发展傅立叶分析的非线性偏微分方程的这一基本领域。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。这些近似的有效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下，这些近似模型的解决方案的定性行为捕获的物理设置的功能。正是在这里，这些解决方案的理论研究变得至关重要。因此，一个开始与特殊的解决方案和它的性质，稳定性，长期行为，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一些数值模拟

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