Regularity for integro-differential operators
Final Report Abstract
In this project, we study equations with discontinuous movements of particles. These types of equations play an important role in several fields such as physics, economics, or biology and the solutions of such equations are given by functions. The equations we are interested in involve integration and differentiation and are called integro-differential equations. They can be described with the help of so-called integro-differential operators. A prominent example of an integro-differential equation is the Boltzmann equation, which is used to describe the dynamics of a dilute gas. The equation involves derivatives in time and space that describe the movement of mass and it involves an object that acts on the velocity and describes the interaction of particles. This object is known to be the sum of an integro-differential operator and an operator of lower order. Such equations are quite involved and in general, solutions cannot be given explicitly. Therefore, it is important to find qualitative properties of solutions. One type of such properties are so-called regularity estimates. They describe whether solutions satisfy certain continuity or differentiability properties. To find such regularity estimates for solutions, we need to derive certain properties of the integro-differential operator. An important condition for integro-differential operators is the so-called coercivity estimate that estimates the corresponding energy form of the operator with an object that naturally appears in the context of fractional derivatives (Sobolev-seminorm of fractional order). In this project, we were seeking for conditions of integro-differential operators so that the corresponding energy form satisfies a coercivity estimate. In a joint work with Prof. Luis Silvestre from the University of Chicago we found a general condition for integro-differential operators that ensures coercivity. This result is applicable to several problems. In particular, it applies to the integro-differential operator appearing in the Boltzmann equation under certain macroscopic bounds of the solution. Another aim of this project was to prove regularity estimates for solutions to equations where the underlying integro-differential operator is anisotropic and singular. The operators under consideration have several fractional orders of differentiability that depend on the respective direction. Furthermore, the operator only charges differences that happen in the directions of the coordinate axes. In a joint project with Prof. Moritz Kassmann and Marvin Weidner from Bielefeld University, we studied a large class of integro-differential operators that allow for such anisotropic and singular behavior of the operator but are not restricted to it. We proved regularity estimates for solutions to corresponding equations. Furthermore, we constructed several examples which show that our theory is not restricted to anisotropic and singular operators and we can even study isotropic and much less singular integro-differential operators. One feature of this work is that all results are robust. This means that they recover classical (local) results by considering the limit of the order of differentiability.
Publications
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Robust Hölder Estimates for Parabolic Nonlocal Operators. 2019
J. Chaker and M. Kassmann and M. Weidner
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Coercivity estimates for integro-differential operators, Calc. Var. Partial Differential Equations 59(4):20, Id/ No 106, 2020
J. Chaker and L. Silvestre
