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Solving partial differential equations and systems by techniques of harmonic analysis

通过调和分析技术求解偏微分方程和系统

基本信息

批准号：
EP/F014589/1
负责人：
Martin Dindos
金额：
$ 36.46万
依托单位：
University of Edinburgh
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2007
资助国家：
英国
起止时间：
2007 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FF014589%2F1
关键词：
Solving partial differential equations systems

项目摘要

The proposed research aims to study important classes of elliptic partial differential equations. Partial differential equations are used to mathematically describe behaviour of many real life phenomena and arise practically everywhere. In our research we plan to focus on a special class of such equations - elliptic. Equations of this type can be encountered in physics, material science, geometry, probability and many other disciplines. In many real life applications, the equations that arise have certain singularities. For example the domain of equation can have corners, cusps or the coefficients of equation itself might be discontinuous. Here the discontinuity of coefficients is the mathematical expression of the fact that many materials contain impurities (foreign objects) that somewhat change the properties of studied objects. For these reasons it is very important to consider these situations mathematically. One particular example of an important elliptic system is the stationary Navier-Stokes equation that arises in mathematical physics (fluid flow). To nonspecialist this equation might look simple, however mathematically it is extremely challenging and our understanding of it is very incomplete. One particular question that remains open is the global existence of smooth solutions of this equation for arbitrary large initial data. We plan to look at related problem - global existence of solutions for the stationary Navier-Stokes equation. The word stationary means that we look for solutions that do not change in time. This assumption makes the equation elliptic and therefore approachable by methods of harmonic analysis we plan to use.

本研究旨在研究一类重要的椭圆型偏微分方程。偏微分方程用于数学上描述许多现实生活现象的行为，并且几乎无处不在。在我们的研究中，我们计划集中研究一类特殊的此类方程——椭圆方程。这种类型的方程可以在物理、材料科学、几何、概率和许多其他学科中遇到。在许多实际应用中，出现的方程具有一定的奇异性。例如，方程的定域可以有角点、尖点，或者方程本身的系数可能是不连续的。这里，系数的不连续是许多材料含有杂质（异物）这一事实的数学表达式，这些杂质会在一定程度上改变所研究对象的性质。由于这些原因，从数学上考虑这些情况是非常重要的。一个重要的椭圆系统的特殊例子是在数学物理（流体流动）中出现的平稳Navier-Stokes方程。对于非专业人士来说，这个方程可能看起来很简单，但在数学上它极具挑战性，我们对它的理解非常不完整。对于任意大的初始数据，该方程的光滑解的全局存在性是一个有待解决的问题。我们计划研究相关问题——平稳Navier-Stokes方程解的整体存在性。平稳这个词的意思是我们寻找不随时间变化的解。这个假设使得方程是椭圆的，因此可以用调和分析的方法来逼近。