Regularity for integro-differential operators

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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).