A heterogeneous nonlocal advection--diffusion system

Abstract

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 $\nabla K_{ij} \in L^{q_{ij}}(\mathbb{R}^d),\, 1 < q_{ij} < \infty$. Local well-posedness of the mild solution and its regularity is obtained using semigroup theory and contraction mapping arguments. For families of kernels that satisfy a given interaction cycle condition, global existence is established using a Nash-type inequality to show an a priori energy bound. For a separate class of kernels that need not satisfy the interaction cycle condition, a smallness condition on the initial data is provided for a uniform-in-time bound. Numerical examples are then considered to illustrate the influence of the kernel regularity on the solutions.