Nonlocal-to-local convergence of convolution operators and some applications

The goal of nonlocal-to-local convergence is to show that certain singular, nonlocal convolution-type integral operators converge to a local differential operator as the convolution kernel concentrates at zero. This can be a useful tool in the physical justification of mathematical models (e.g., the Cahn-Hilliard equation), especially when a desired local differential operator cannot be derived by microscopic laws.
The nonlocal-to-local convergence of convolution operators with radially symmetric (i.e., isotropic) kernels having $W^{1,1}$-regularity is already very well understood. We discuss the Cahn-Hilliard equation as well as a Navier-Stokes-Cahn-Hilliard model as possible applications. However, the assumption of $W^{1,1}$-regularity is too strong for many applications. Also, in some situations (e.g., crystallization phenomena), convolution kernels are not radially symmetric but merely even (i.e., anisotropic).

In an ongoing collaboration (joint work with Helmut Abels and Christoph Hurm), we intend to establish strong nonlocal-to-local convergence results with convergence rates for anisotropic kernels satisfying lower regularity assumptions.

These results can, for example, be applied to nonlocal phase-field models such as the anisotropic Cahn-Hilliard equation.