Regularity for integro-differential operators

这个研究项目的目的是为控制粒子不连续运动的方程发展一种新的数学理论。这类方程在一些科学领域，如数学、物理和生物学中，具有重要的基础意义。自1872年以来，所谓的玻尔兹曼方程是描述介质（例如气体）中粒子的基本模型。气体运动论假定气体是由大量的粒子组成的，并通过粒子的运动来描述气体的性质。该理论假定粒子不断运动并相互碰撞。这样的物理系统可以用方程进行数学建模，这些方程的解是依赖于模型物理量的函数。对于玻尔兹曼方程，未知函数描述了粒子在介质中的统计分布，作为时间、空间和速度的函数。该方程是所谓的偏积分微分方程（PIDE），其中分布由两个数学算子描述。一方面，所需要的函数是通过其环境中的值的导数来描述的。另一方面，由于积分算子，考虑了整个空间上的函数值（非局部）。在玻尔兹曼方程中，这种非定域性通过粒子的碰撞发生。在这个项目中，我们研究了一类更广泛的动力学PIDE，其中玻尔兹曼方程作为特殊情况出现。该方程通过考虑一类更大的非局部碰撞来推广，这些碰撞可以用所谓的非局部算子在数学上描述，由一类可测量的积分核决定。我们研究了碰撞项的性质和PIDE的解。这些解只能在非常严格的例外情况下显式计算。因此，确定这种溶液的定性性质是至关重要的。函数的一个重要特征是连续导数的个数，从而可以赋值到一定的函数空间。本课题旨在证明在水动力量的某些有界性质下，解是无穷多次连续可微的。另一个目的是证明给定的一类积分核在PIDE中的非局部表达式与分数阶导数理论（Sobolev-Slobodeckij半模）中出现的已知对象具有可比性。此外，我们的目标是在证明奇异非局部算子的正则性理论（Aleksandrov-Bakelman-Pucci极大原理）的理论工具方面取得进展。

The aim of this research project is to develop a new mathematical theory for equations which govern the discontinuous motion of particles. Such equations are of fundamental importance in several areas of science, e.g., in Mathematical Physics and Biology. Since 1872, the so called Boltzmann equation is the base model for the description of particles in a medium (for instance a gas). The kinetic theory of gases assumes that gases consist of a large number of particles and describes properties of gases through the motions of the particles. The theory assumes that the particles are constantly in motion and collide with each other. Such physical systems can be mathematically modeled by equations whose solutions are functions depending on the physical quantities of the model. For the Boltzmann equation, the unknown function describes the statistical distribution of particles in a medium as a function of time, space and velocity. The equation is a so-called partial integro-differential equation (PIDE) in which the distribution is described by two mathematical operators. On the one hand, the required function is described through derivatives by values ​​in its environment. On the other hand, due to the integral operators, function values ​on the whole space ​are taken into account (nonlocal). In the Boltzmann equation, this nonlocality occurs through collisions of the particles. Within this project we research on a wider class of kinetic PIDE for which the Boltzmann equation occurs as a special case. The equation is generalized by considering a larger class of nonlocal collisions, which can be mathematically described by so-called nonlocal operators, determined by a class of measurable integration kernels. We investigate properties of the collision term and solutions to PIDE. These solutions can only be explicitly calculated in very restrictive exceptional cases. Therefore, it is crucial to determine qualitative properties of such solutions. An important characteristic of functions is the number of continuous derivatives and thereby the assignment into certain function spaces. This research project intends to prove that, under some boundedness properties on the hydrodynamic quantities, solutions are infinitely many times continuously differentiable. Another aim is to prove that the nonlocal expression in the PIDE for a given class of integration kernels is comparable to known objects occurring in the theory of fractional order derivatives (Sobolev-Slobodeckij seminorm). Furthermore, we aim to make progress towards the proof of a theoretical tool in the regularity theory of singular nonlocal operators (Aleksandrov-Bakelman-Pucci Maximum Principle).