A heterogeneous nonlocal advection--diffusion system

We present a self-contained investigation on the local and global well-posedness for a system of nonlocal advection–diffusion equations for a heterogeneous population over ${\mathbb{R}}^d$, $d\in \mathbb{N}$. Each convolution kernel $K_{ij}$, which describes the nonlocal advection of species $i$ according to the distribution of species $j$, is assumed to have its own regularity $\mathrm{\nabla }{K}_{ij}\in L^{q_{ij}}({\mathbb{R}}^d),1<q_{ij}<\mathrm{\infty }$. Local well-posedness of the mild solution and its regularity is obtained using semigroup theory and contraction mapping arguments. For families of kernels classified as regular, a global bound is established using a Nash-type inequality. For suitable irregular families of kernels, global boundedness is instead obtained via a smallness condition on the initial data. A one-dimensional numerical example is provided to illustrate the influence of kernel regularity on the solutions.