A local sensitivity analysis for the kinetic Cucker-Smale equation with random inputs
type: Item
instance of: scholarly article
title: A local sensitivity analysis for the kinetic Cucker-Smale equation with random inputs
zbMATH Open document ID: 1395.35178
author: Seung-Yeal Ha
author: Jinwook Jung
author: Shi Jin
published in: Journal of Differential Equations
publication date: 2018-07-11T00:00:00Z
review text: The authors consider the kinetic Cucker-Smale problem with random inputs \( \partial _{t}f+v\cdot \nabla _{x}f+\nabla _{v}(F_{a}[f])=0\), for \(x,v\in \mathbb{R}^{d}\), \(z\in \Omega \) the 1D random space, \(t>0\), where \( F_{a}[f](t,x,v,z)=-\int_{\mathbb{R}^{2d}}\psi (x-x_{\ast },z)(v-v_{\ast })f(t,x_{\ast },v_{\ast },z)dv_{\ast }dx_{\ast }\). The function \(\psi (x,z)= \widetilde{\psi }(\left| x\right| ,z)\) is such that there exists a positive random variable \(\psi _{M}(z)\) satisfying \(0<\psi (x,z)\leq \psi _{M}(z)<\infty \) and satisfies \(\psi (-x,z)=\psi (x,z)\), \((\widetilde{\psi } (\left| x_{2}\right| ,z)-\widetilde{\psi }(\left| x_{1}\right| ,z))(\left| x_{2}\right| -\left| x_{1}\right| )\leq 0\), \(\widetilde{\psi }(\cdot ,z)\in Lip(\mathbb{R}; \mathbb{R}_{+})\). The solution \(f\) starts from an initial data \(f^{0}(x,v,z)\) at \(t=0\). The paper mainly analyzes the effects of randomness of the above Cucker-Smale problem on flocking dynamics and on the regularity of the solution, through a local sensitivity analysis. The first main result proves the existence of a solution \(f\in W^{k,\infty }(\mathbb{R}^{2d}\times [ 0,T))\), for every \(T>0\), assuming that \(\psi \in L^{\infty }(\Omega ;W^{k,\infty }(\mathbb{R}^{d}))\) and \(f^{0}\) is compactly supported and \( W^{k,\infty }\)-regular. For the proof, the authors introduce the transport operator \(\mathcal{T}=\partial _{t}+v\cdot \nabla _{x}+F_{a}[f]\cdot \nabla _{v}\) associated to the problem and they follow lines of the paper by \textit{S. Y. Ha} and \textit{J. G. Liu} in [Kinet. Relat. Models 1, No. 3, 415--435 (2008; Zbl 1402.76108)] in the case \(k=1\), to prove the existence result through Gronwall's Lemma. The second main result of the paper presents a pathwise flocking estimate. The authors define the mean bulk velocity \(v_{c}(t,z)=\frac{1}{\mathcal{M}_{0}(z)}\int_{\mathbb{R} ^{2d}}vf(t,x,v,z)dvdx\) where \(\mathcal{M}_{0}(z)=\int_{\mathbb{R} ^{2d}}f(t,x,v,z)dvdx\). They also define the …
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 35Q82
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35Q92
Mathematics Subject Classification ID: 37H99
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 6901114
zbMATH Keywords: Cucker-Smale model
zbMATH Keywords: random inputs
zbMATH Keywords: pathwise flocking estimate
zbMATH Keywords: local sensitivity analysis
zbMATH Keywords: random communication
zbMATH Keywords: uncertainty quantification
MaRDI profile type: Publication
full work available at URL: j.jde.2018.05.013
OpenAlex ID: W2878053916
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Wikidata QID: Q129542675
DOI: 10.1016/J.JDE.2018.05.013
description: scientific article; zbMATH DE number 6901114