Gradient regularity in nonlinear nonlocal PDEs

Classical (local) partial differential equations describe phenomena where the change of a function is determined only by the state at a point or its immediate neighborhood. An example of this is the heat conduction equation: it describes how the temperature at a point in space changes over time and depends only on the temperature in the immediate neighborhood. In contrast, in nonlocal partial differential equations, the value of the function at a specific point depends not only on the values and derivatives of the function in the immediate neighborhood of that point, but also on information from points that are far away. This allows for the modeling of phenomena that exhibit "jump-like" or "anomalous" dynamics. Such phenomena occur in many different fields, such as financial markets, anomalous diffusion, biological processes, communication networks, and geophysics. The nonlocal differential equations considered in this project are implemented using fractional derivatives. Fractional derivatives provide a way to extend the concept of the classical derivative so that it can also be defined for non-integer exponents such as 1/2 or 1/3. In the differential equations, fractional derivatives now replace the classical derivatives. In this project, we aim to better understand the solutions to such differential equations, particularly their regularity. Although the field is currently very popular, there are still many open and challenging questions, especially in the area of nonlinear nonlocal partial differential equations. For example, the optimal regularity of solutions is not known. The time-dependent variant is almost entirely unexplored. The methods used to solve such problems are diverse. They involve deep knowledge of real analysis, functional analysis, function spaces such as fractional Sobolev and Nikolskii spaces, and the theory of nonlinear partial differential equations. One difficulty in studying nonlocal differential equations is that even if one is only interested in a small region of space, the behavior of the solution over the entire space must always be considered. The goal of this project is to develop new methods that allow for a better understanding of the regularity of solutions.

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