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Keller-Segel type approximation for nonlocal Fokker-Planck equations in one-dimensional bounded domain

Hideki Murakawa, Yoshitaro Tanaka

Abstract

Numerous evolution equations with nonlocal convolution-type interactions have been proposed. In some cases, a convolution was imposed as the velocity in the advection term. Motivated by analyzing these equations, we approximate advective nonlocal interactions as local ones, thereby converting the effect of nonlocality. In this study, we investigate whether the solution to the nonlocal Fokker-Planck equation can be approximated using the Keller-Segel system. By singular limit analysis, we show that this approximation is feasible for the Fokker-Planck equation with any potential and that the convergence rate is specified. Moreover, we provide an explicit formula for determining the coefficient of the Lagrange interpolation polynomial with Chebyshev nodes. Using this formula, the Keller-Segel system parameters for the approximation are explicitly specified by the shape of the potential in the Fokker-Planck equation. Consequently, we demonstrate the relationship between advective nonlocal interactions and a local dynamical system.

Keller-Segel type approximation for nonlocal Fokker-Planck equations in one-dimensional bounded domain

Abstract

Numerous evolution equations with nonlocal convolution-type interactions have been proposed. In some cases, a convolution was imposed as the velocity in the advection term. Motivated by analyzing these equations, we approximate advective nonlocal interactions as local ones, thereby converting the effect of nonlocality. In this study, we investigate whether the solution to the nonlocal Fokker-Planck equation can be approximated using the Keller-Segel system. By singular limit analysis, we show that this approximation is feasible for the Fokker-Planck equation with any potential and that the convergence rate is specified. Moreover, we provide an explicit formula for determining the coefficient of the Lagrange interpolation polynomial with Chebyshev nodes. Using this formula, the Keller-Segel system parameters for the approximation are explicitly specified by the shape of the potential in the Fokker-Planck equation. Consequently, we demonstrate the relationship between advective nonlocal interactions and a local dynamical system.
Paper Structure (14 sections, 27 theorems, 193 equations, 8 figures)

This paper contains 14 sections, 27 theorems, 193 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Mathematical settings and main results
  3. Mild solution
  4. Fundamental solution and boundedness
  5. Contraction map
  6. Singular limit analysis
  7. Fundamental solution
  8. Boundedness of auxiliary factors
  9. Order estimation
  10. Coefficients of linear sum
  11. Linear stability analysis
  12. Concluding remarks
  13. Proof of Lemma \ref{['lemm:bounded_G']}
  14. Proof of Lemma \ref{['lemm:k_j_L^1']}

Key Result

Proposition 2.1

Let $R>0$ be an arbitrary real number, and assume that $W \in W_{\mathrm{per}}^{1,1}(\Omega):=\{u|_\Omega \in W^{1,1}(\Omega) \ | \ u(x)=u(x+2L), \ x \in \mathbb{R} \}$ and Then for any $T>0$ there exists a unique mild solution $\rho$ to nonlocal:FP in $C( [0, T], H^1(\Omega) ) \cap L^2( 0, T, H^2(\Omega) )$ satisfying where $C_0$ depends on $R$ and $T$. Moreover, this mild solution satisfies no

Figures (8)

  • Figure 1: Results of a numerical simulation of the approximation for $W$ by the linear combination of $\cosh j(L-|x|)$. We set $W(x)=e^{-5x^2}(\cos(3\pi) x- 1/2 \cos(2\pi x))$, and $L=2$. (a) Profiles of $W$ and the linear sum of $\cosh j(L-|x|)$. (b) Profiles of $f$ and the Lagrange interpolation polynomial on $[1, \cosh L]$. (c) Distribution of $\{ \alpha^9_j \}_{j=0}^9$.
  • Figure 2: Results of numerical simulations for \ref{['nonlocal:FP']} with a potential $W(x) = e^{-5x^2}$ and $\mu$ defined in \ref{['NFP:mu']}, and \ref{['eq:system']} with $M=7$. The parameters are given by $L=1$, $\varepsilon=0.001$, $d_1=1000000$, $\mu=5$ and $d_j$ and $a_j$ are provided by \ref{['set:dj']} and \ref{['set:aj']}, respectively. (a) Profiles of the numerical result of \ref{['nonlocal:FP']} at $t=20.0$. The horizontal and vertical axes correspond to the position $x$ and $\rho$, respectively. The red curve is the numerical result of $\rho$. (b) Profiles of the numerical result of \ref{['eq:system']} at $t=20.0$. We impose the same initial data for $\rho^\varepsilon$ as that of $\rho$ and $(v_j)_0= k_j*\rho_0, \ (j=1,\ldots,M)$. The axes are set same as that of (a). The red and the other color curves correspond to $(\rho^\varepsilon,\{v_j^\varepsilon\}_{j=1}^7)$, respectively. (c) Profiles of $W$ and $\sum_{j=0}^6 \alpha_j^6 \cosh(j(L-|x|))$. The orange dashed and blue curves corresponding to $W$ and $\sum_{j=0}^6 \alpha_j^6 \cosh(j(L-|x|))$, respectively are drawn in a same plane. (d) The distribution of $\{\alpha_j^6\}_{j=0}^6$.
  • Figure 3: Results of a numerical simulation for \ref{['NFP:mu']} with \ref{['kernel:lali']}. The parameters are $\mu=5.0$, $d_1=0.1$, and $d_2=3.0$ and the initial datum are given by $1.0$ with small perturbations. The horizontal and vertical axes correspond to the position $x$ and value of solution $\rho$, respectively. The red curve corresponds to the solution $\rho$. The left, middle left, middle right and right pictures exhibit the profiles of solutions of \ref{['NFP:mu']} with \ref{['kernel:lali']} in the interval $[0, 10]$ at $t = 0, 0.5, 1.0$ and $3.0$, respectively.
  • Figure 4: Profile of the integral kernel \ref{['kernel:lali']}, and distribution of eigenvalue with $\omega_1(n)$ with same parameters as those in Fig \ref{['fig1']}.
  • Figure 5: Results of a numerical simulation for \ref{['NFP:mu']} with \ref{['pot:attract']}. The parameters are $\mu=4.0$ and $R=1.0$ and the initial datum are given by $1.0$ with small perturbations. The horizontal and vertical axes correspond to the position $x$ and value of solution $\rho$, respectively. The red curve corresponds to the solution $\rho$. The left, middle left, middle right and right pictures exhibit the profiles of solutions of \ref{['NFP:mu']} with \ref{['pot:attract']} in the interval $[0, 10]$ at $t = 0, 0.8, 2.0$ and $12.0$, respectively.
  • ...and 3 more figures

Theorems & Definitions (53)

  • Proposition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Remark 2.5
  • Corollary 2.6
  • Lemma 2.7
  • Remark 2.8
  • Theorem 2.9
  • Remark 2.10
  • ...and 43 more