# Diffusive and Anti-Diffusive Behavior for Kinetic Models of Opinion Dynamics

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Case**

**1.**

**Case**

**2.**

**Case**

**3.**

**Proposition**

**1.**

- 1.
- If $\sigma =-1$ the solution is global and possesses all finite ${L}^{p}$-norms, $p>1$, and the functions $t\mapsto {\parallel f\left(t\right)\parallel}_{p}$ are decreasing;
- 2.
- If $\sigma =1$ the solution, depending on initial data, is either global ($T=\infty $) or local ($T<\infty $), it possesses all finite ${L}^{p}$-norms on $[0,T)$, $p>1$, and the functions $t\mapsto {\parallel f\left(t\right)\parallel}_{p}$ are increasing for $t\in [0,T)$.

**Proof.**

**Proposition**

**2.**

**Proof.**

## 2. Formal Diffusive and Anti-Diffusive Limits

**Assumption**

**1.**

- 1.
- The function $\overline{\beta}:{\mathbb{R}}^{d}\to \mathbb{R}$ is bounded, non-negative and symmetric, i.e., $\overline{\beta}\left(v\right)=\overline{\beta}(-v)\phantom{\rule{0.166667em}{0ex}}$;
- 2.
- The first, second, third, and fourth generalized moments are bounded, i.e.,$$\underset{{\mathbb{R}}^{d}}{\int}{\left|v\right|}^{p}\phantom{\rule{0.166667em}{0ex}}\overline{\beta}\left(v\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}v<+\infty \phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}p=1,2,3,4\phantom{\rule{0.166667em}{0ex}};$$
- 3.
- The matrix$${\Sigma}_{\overline{\beta}}=\left[\phantom{\rule{0.277778em}{0ex}}\underset{{\mathbb{R}}^{d}}{\int}{v}_{i}\phantom{\rule{0.166667em}{0ex}}{v}_{j}\phantom{\rule{0.166667em}{0ex}}\overline{\beta}\left(v\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}v\phantom{\rule{0.277778em}{0ex}}\right]\phantom{\rule{0.166667em}{0ex}},$$

## 3. Convergence Result

**Proposition**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

- the kinetic equation is transformed into an equation formally equivalent to a porous-media equation with a small perturbation;
- the perturbed porous-media equation is reduced to its standard form by the change of variables $u\to Bu$;
- the comparison principle for porous-media equations is applied.

**Corollary**

**1.**

**Proof.**

## 4. Numerical Examples

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Parisot, M.; Lachowicz, M. A kinetic model for the formation of swarms with nonlinear interactions. Kinet. Relat. Model.
**2016**, 9, 131–164. [Google Scholar] [CrossRef] - Lachowicz, M.; Leszczyński, H.; Parisot, M. A simple kinetic equation of swarm formation: Blow–up and global existence. Appl. Math. Lett.
**2016**, 57, 104–107. [Google Scholar] [CrossRef] - Lachowicz, M.; Leszczyński, H.; Parisot, M. Blow–up and global existence for a kinetic equation of swarm formation. Math. Model. Methods Appl. Sci.
**2017**, 27, 1153–1175. [Google Scholar] [CrossRef] - Lachowicz, M.; Leszczyński, H.; Topolski, K.A. Self–organization with small range interactions: Equilibria and creation of bipolarity. Appl. Math. Comput.
**2019**, 343, 156–166. [Google Scholar] [CrossRef] - Marsan, G.A.; Bellomo, N.; Gibelli, L. Stochastic evolutionary differential games toward a system theory of behavioral social dynamics. Math. Model. Methods Appl. Sci.
**2016**, 26, 1051–1093. [Google Scholar] [CrossRef] - Banasiak, J.; Lachowicz, M. Methods of Small Parameter in Mathematical Biology; Birkhäuser: Boston, MA, USA, 2014. [Google Scholar]
- Lachowicz, M. Individually–based Markov processes modeling nonlinear systems in mathematical biology. Nonlinear Anal. Real World Appl.
**2011**, 12, 2396–2407. [Google Scholar] [CrossRef] - Banasiak, J.; Lachowicz, M. On a macroscopic limit of a kinetic model of alignment. Math. Model. Methods Appl. Sci.
**2013**, 23, 2647–2670. [Google Scholar] [CrossRef] - Bellomo, N.; Soler, J. On the mathematical theory of the dynamics of swarms viewed as complex systems. Math. Model. Methods Appl. Sci.
**2012**, 22, 1140006. [Google Scholar] [CrossRef] - Carrillo, J.A.; Fornasier, M.; Toscani, G.; Vecil, F. Particle, Kinetic, and Hydrodynamic Models of Swarming. In Mathematical Modeling of Collective Behavior in Socio–Economic and Life Sciences; Naldi, G., Pareschi, L., Toscani, G., Eds.; Birkhäuser: Boston, MA, USA, 2010; pp. 297–336. [Google Scholar]
- Carrillo, J.A.; D’Orsogna, M.R.; Panferov, V. Double milling in self–propelled swarms from kinetic theory. Kinet. Relat. Model.
**2009**, 2, 363–378. [Google Scholar] [CrossRef] - Carlen, E.; Degond, P.; Wennberg, B. Kinetic limits for pair–interaction driven master equation and biological swarm models. Math. Model. Methods Appl. Sci.
**2013**, 23, 1339–1376. [Google Scholar] [CrossRef] - Degond, P.; Frouvelle, A.; Raoul, G. Local stability of perfect alignment for a spatially homogeneous kinetic model. J. Stat. Phys.
**2014**, 157, 84–112. [Google Scholar] [CrossRef] - Aronson, D.G. The porous medium equation. In Nonlinear Diffusion Problems; Fasano, A., Primicerio, M., Eds.; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 1986; Volume 1224, pp. 1–46. [Google Scholar]
- Vázquez, J.J. The Porous Medium Equation: Mathematical Theory; Oxford University Press: Oxford, UK, 2007. [Google Scholar]
- Bénilan, P.; Crandall, M.G.; Pierre, M. Solutions of porous medium equation in ${\mathbb{R}}^{N}$ under optimal conditions on initial values. Indiana Univ. Math. J.
**1984**, 33, 51–87. [Google Scholar] [CrossRef] - Lachowicz, M.; Wrzosek, D. Nonlocal bilinear equations: Equilibrium solutions and diffusive limit. Math. Model. Methods Appl. Sci.
**2001**, 11, 1393–1409. [Google Scholar] [CrossRef] - Benilan, P.; Brezis, H.; Crandall, M.G. A semilinear equation in L
_{1}(${\mathbb{R}}^{N}$). Ann. Scuola Norm. Sup. Pisa**1975**, 2, 523–555. [Google Scholar] - Barbu, V. Nonlinear Semigroups and Differential Equations in Banach Spaces; Noordhoff: Leyden, IL, USA, 1976. [Google Scholar]

**Figure 1.**Initial function ${f}_{0}\left(u\right)=Cexp\left(\frac{-1}{1-{u}^{2}}\right)$ for $u\in (-1,1)$.

**Figure 2.**Time derivative ${\partial}_{t}f(t,u)$ for $t=0$, $\sigma =-1$ with ${f}_{0}\left(u\right)=Cexp\left(\frac{-1}{1-{u}^{2}}\right)$ for $u\in (-1,1)$.

**Figure 3.**Method of lines for $\epsilon ={10}^{-1}$, ${f}_{0}\left(u\right)=Cexp\left(\frac{-1}{1-{u}^{2}}\right)$, $\sigma =-1$; $t=0.0$ (blue), $t=0.5$ (red), $t=1.0$ (yellow), $t=2.0$ (green).

**Figure 4.**Time derivative ${\partial}_{t}f(t,u)$ for $t=0$, $\sigma =1$ with ${f}_{0}\left(u\right)=Cexp\left(\frac{-1}{1-{u}^{2}}\right)$ for $u\in (-1,1)$.

**Figure 5.**Method of lines for $\epsilon ={10}^{-1}$, ${f}_{0}\left(u\right)=Cexp\left(\frac{-1}{1-{u}^{2}}\right)$, $\sigma =1$; $t=0.0$ (blue), $t=0.2$ (red).

**Figure 6.**Initial function ${f}_{0}\left(x\right)=C(1+{u}^{2})exp\left(\frac{-1}{1-{u}^{2}}\right)$ for $u\in (-1,1)$.

**Figure 7.**Derivative ${f}_{0}^{\u2033}$ for ${f}_{0}\left(u\right)=C(1+{u}^{2})exp\left(\frac{-1}{1-{u}^{2}}\right)$ for $u\in (-1,1)$.

**Figure 8.**Derivative ${\partial}_{t}f(t,u)$ for $t=0$, $\sigma =1$ with ${f}_{0}\left(u\right)=C(1+{u}^{2})exp\left(\frac{-1}{1-{u}^{2}}\right)$ for $u\in (-1,1)$.

**Figure 9.**Method of lines for $\epsilon ={10}^{-1}$, ${f}_{0}\left(u\right)=C(1+{u}^{2})exp\left(\frac{-1}{1-{u}^{2}}\right)$, $\sigma =1$; $t=0.0$ (blue), $t=0.2$ (red), $t=0.24$ (yellow).

**Figure 10.**Derivative ${\partial}_{t}f(t,u)$ for $t=0$, $\sigma =-1$ with ${f}_{0}\left(u\right)=C(1+{u}^{2})exp\left(\frac{-1}{1-{u}^{2}}\right)$ for $u\in (-1,1)$.

**Figure 11.**Method of lines for $\epsilon ={10}^{-1}$, ${f}_{0}\left(u\right)=C(1+{u}^{2})exp\left(\frac{-1}{1-{u}^{2}}\right)$, $\sigma =-1$; $t=0.0$ (blue), $t=0.5$ (red), $t=1.0$ (yellow).

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lachowicz, M.; Leszczyński, H.; Puźniakowska–Gałuch, E.
Diffusive and Anti-Diffusive Behavior for Kinetic Models of Opinion Dynamics. *Symmetry* **2019**, *11*, 1024.
https://doi.org/10.3390/sym11081024

**AMA Style**

Lachowicz M, Leszczyński H, Puźniakowska–Gałuch E.
Diffusive and Anti-Diffusive Behavior for Kinetic Models of Opinion Dynamics. *Symmetry*. 2019; 11(8):1024.
https://doi.org/10.3390/sym11081024

**Chicago/Turabian Style**

Lachowicz, Mirosław, Henryk Leszczyński, and Elżbieta Puźniakowska–Gałuch.
2019. "Diffusive and Anti-Diffusive Behavior for Kinetic Models of Opinion Dynamics" *Symmetry* 11, no. 8: 1024.
https://doi.org/10.3390/sym11081024