Solving partial differential equations and systems by techniques of harmonic analysis

本研究旨在研究一类重要的椭圆型偏微分方程族。偏微分方程被用来用数学来描述许多现实生活现象的行为，几乎随处可见。在我们的研究中，我们计划重点研究一类特殊的此类方程--椭圆型方程。这种类型的方程在物理、材料科学、几何、概率和许多其他学科中都可以遇到。在许多实际应用中，出现的方程都有一定的奇点。例如，方程的区域可以有角、尖点，或者方程本身的系数可能是不连续的。在这里，系数的不连续性是许多材料含有杂质(异物)这一事实的数学表达式，这些杂质在某种程度上改变了所研究对象的性质。出于这些原因，从数学上考虑这些情况是非常重要的。一个重要的椭圆系的特殊例子是在数学物理(流体流动)中出现的定常的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，但在数学上它是极其具有挑战性的，我们对它的理解也非常不完整。一个悬而未决的问题是，对于任意大的初始数据，该方程的整体光滑解的存在性。我们计划研究相关问题-定常的Navier-Stokes方程整体解的存在性。固定这个词的意思是我们寻找不随时间变化的解决方案。这个假设使得方程是椭圆的，因此可以通过我们计划使用的调和分析方法来逼近。

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.