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A heterogeneous nonlocal advection--diffusion system

Joseph McCusker, John Christopher Meyer, Mabel Lizzy Rajendran

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 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.

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 , . Each convolution kernel , which describes the nonlocal advection of species according to the distribution of species , is assumed to have its own regularity . 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.
Paper Structure (11 sections, 37 theorems, 167 equations, 3 figures)

This paper contains 11 sections, 37 theorems, 167 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations and Definitions
  3. Main results and structure of the paper
  4. Preliminaries
  5. Local well-posedness
  6. Regularity of the solution
  7. Non-negativity and conservation of mass
  8. Global in time bound
  9. Dynamical properties
  10. Numerical simulations
  11. Discussion

Key Result

Theorem 3.1

Suppose that $\nabla K \in L^Q(\mathbb{R}^d)$ for some $Q \in (1,\infty]^{N \times N}$ and $u_{0} \in L^1 \cap L^{P}(\mathbb{R}^d)$, for some $P \in [1,\infty)^N$ such that, for all $j \in \{1,\dots,N\}$, Then, there exists some $T > 0$ and a unique $u \in C([0,T]; L^1 \cap L^{P}(\mathbb{R}^d))$ which is a mild solution to eqn:agg_diff up to time $T$. Furthermore, the solution in the $L^1 \cap L^

Figures (3)

  • Figure 1: The constraint graph for the case $N = 3$.
  • Figure 2: Graphs of $W_s(x)$ and $W'_s(x)$, as defined in \ref{['defn:hats']}--\ref{['defn:hat_deriv']}, for various $s$.
  • Figure 3: log--log plots showing the relationship between kernel strengths $\gamma$ and the large time $L^2$ norm of each species in a non-coupled vs coupled system. For both scenarios, the initial data is $u_{0} = (\chi_{[-1,1]},\, \chi_{[-1,1]})$, the diffusion constants are $D_1 = D_2 = 0.1$ and $s_1 = 15/4,\, s_2 = 12/5$. Numerical convergence was observed for all plotted data points and the $L^2$ norms stabilised very quickly, i.e. for $t$ less than $10$.

Theorems & Definitions (77)

  • Definition 2.1: Heat semigroup
  • Definition 2.2: Mild solution
  • Definition 2.3: Interaction cycle
  • Remark 2.4
  • Theorem 3.1: Local well-posedness
  • Theorem 3.2: Regularity, non-negativity and conservation of mass
  • Theorem 3.3: Global existence for regular systems
  • Theorem 3.4: Smallness condition for irregular systems
  • Lemma 4.1: evans2010
  • Remark 4.2
  • ...and 67 more