A heterogeneous nonlocal advection--diffusion system
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911525882560512 |
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| author | McCusker, Joseph Meyer, John Christopher Rajendran, Mabel Lizzy |
| author_facet | McCusker, Joseph Meyer, John Christopher Rajendran, Mabel Lizzy |
| contents | 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17749 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A heterogeneous nonlocal advection--diffusion system McCusker, Joseph Meyer, John Christopher Rajendran, Mabel Lizzy Analysis of PDEs 35A01, 35B45, 35R09, 35K40, 05C90 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 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. |
| title | A heterogeneous nonlocal advection--diffusion system |
| topic | Analysis of PDEs 35A01, 35B45, 35R09, 35K40, 05C90 |
| url | https://arxiv.org/abs/2603.17749 |