A local sensitivity analysis for the kinetic Cucker-Smale equation with random inputs
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 Lyapunov functional \(\mathcal{L} [f](t,z)=\int_{\mathbb{R}^{2d}}\left| v-v_{c}(z)\right| ^{2}f(t,x,v,z)dvdx\). The authors prove an exponential decay of \(\mathcal{L}\) , assuming that \(f\) is a global solution to the above problem and \(f^{0}\) is compactly supported: \(\mathcal{L}[f](t,z)\leq \mathcal{L}[f^{0}](z)\exp (-2 \mathcal{M}_{0}(z)\int_{0}^{t}\varphi (s,z)ds)\), where the expression of \( \varphi \) is given in terms of \(\widetilde{\psi }\) and of data of the problem. The proof is obtained estimating the derivative of \(\mathcal{L} [f](t,z)\) using the Cucker-Smale equation and Gronwall's Lemma. The last main result proves two local sensitivity analyses for a regular solution. The authors observe that the Cucker-Smale equation implies \( \partial _{t}(\partial _{z}^{m}f)+v\cdot \nabla _{x}(\partial _{z}^{m}f)+\sum_{l\leq m}\begin{pmatrix} m \\ l \end{pmatrix} \nabla _{v}(\partial _{z}^{l}F_{a}[f]\partial _{z}^{m-l}f)=0\), for every \(m\in \mathbb{Z}_{+}\). The authors prove the existence of a unique \( W^{k-1,\infty }\)-regular solution process \(\partial _{z}f(z)\in W^{k-1,\infty }(\mathbb{R}^{2d}\times [ 0,T))\), for every \(T>0\), to this last problem, assuming that \(f^{0}\) is compactly supported and satisfies \(f^{0}\left( z)\right) \) and \(\partial _{z}f^{0}(z)\) are \( W^{k,\infty }\)-regular and \(W^{k-1,\infty }\)-regular respectively and \(\psi \in W^{1,\infty }(\Omega ;W^{k,\infty }(\mathbb{R}^{d}))\). The proof follows similar lines to the previous existence result. Then the authors generalize this result to higher-order regularity by induction on \(m\). The paper ends with two pathwise stability results. The authors first prove the existence of a nonnegative random variable \(C(T,z)\) for each \(z\in \Omega \) such that \( \left\| (f-\widetilde{f})(t,z)\right\| _{W_{x,v}^{k,\infty }}\leq C(T,z)\left\| (f^{0}-\widetilde{f}^{0})(z)\right\| _{W_{x,v}^{k,\infty }}\) for any \(T>0\) and every \(t\in (0,T)\), assuming compactly supported and \( W^{k+1,\infty }\)-regularity properties for the initial data \(f^{0}\), \( \widetilde{f}^{0}\) and a \(W^{k+1,\infty }\)-regularity property for the communication weight \(\psi \). The authors finally generalize this stability result to higher orders.
scientific article; zbMATH DE number 6901114
scientific article