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Kinetic & Related Models

October 2020 , Volume 13 , Issue 5

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Kinetic formulation of a 2 × 2 hyperbolic system arising in gas chromatography
Christian Bourdarias, Marguerite Gisclon and Stéphane Junca
2020, 13(5): 869-888 doi: 10.3934/krm.2020030 +[Abstract](285) +[HTML](46) +[PDF](793.17KB)
Abstract:

A particular 2x2 hyperbolic system commonly used in the context of gas-solid chromatography is reformulated as a single kinetic equation using an additional kinetic variable. A kinetic numerical scheme is built from this new formulation and its behavior is tested on solving the Riemann problem in different configurations leading to single or composite waves.

Quantitative local sensitivity estimates for the random kinetic Cucker-Smale model with chemotactic movement
Seung-Yeal Ha and Bora Moon
2020, 13(5): 889-931 doi: 10.3934/krm.2020031 +[Abstract](293) +[HTML](60) +[PDF](641.28KB)
Abstract:

In this paper, we present quantitative local sensitivity estimates for the kinetic chemotaxis Cucker-Smale(CCS) equation with random inputs. In the absence of random inputs, the kinetic CCS model exhibits velocity alignment under suitable structural assumptions on the turning kernel and reaction term despite of the random effect due to a turning operator. We provide a global existence of a regular solution with slow velocity alignment for the random kinetic CCS model within the proposed framework. Moreover, we investigate the propagation of regularity and stability of infinitesimal variations in random space.

On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states
Gerhard Rein and Christopher Straub
2020, 13(5): 933-949 doi: 10.3934/krm.2020032 +[Abstract](284) +[HTML](41) +[PDF](404.21KB)
Abstract:

If the Vlasov-Poisson or Einstein-Vlasov system is linearized about an isotropic steady state, a linear operator arises the properties of which are relevant in the linear as well as nonlinear stability analysis of the given steady state. We prove that when defined on a suitable Hilbert space and equipped with the proper domain of definition this transport operator \begin{document}$ {\mathcal T} $\end{document} is skew-adjoint, i.e., \begin{document}$ {\mathcal T}^\ast = - {\mathcal T} $\end{document}. In the Vlasov-Poisson case we also determine the kernel of this operator.

Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules
Yoshinori Morimoto and Chao-Jiang Xu
2020, 13(5): 951-978 doi: 10.3934/krm.2020033 +[Abstract](281) +[HTML](48) +[PDF](503.07KB)
Abstract:

We consider the Cauchy problem of the nonlinear Landau equation of Maxwellian molecules, under the perturbation frame work to global equilibrium. We show that if \begin{document}$ H^r_x(L^2_v), r >3/2 $\end{document} norm of the initial perturbation is small enough, then the Cauchy problem of the nonlinear Landau equation admits a unique global solution which becomes analytic with respect to both position \begin{document}$ x $\end{document} and velocity \begin{document}$ v $\end{document} variables for any time \begin{document}$ t>0 $\end{document}. This is the first result of analytic smoothing effect for the spatially inhomogeneous nonlinear kinetic equation. The method used here is microlocal analysis and energy estimates. The key point is adopting a time integral weight of exponential type associated with the kinetic transport operator.

On the generic complete synchronization of the discrete Kuramoto model
Woojoo Shim
2020, 13(5): 979-1005 doi: 10.3934/krm.2020034 +[Abstract](271) +[HTML](49) +[PDF](527.05KB)
Abstract:

We study the emergent behavior of discrete-time approximation of the finite-dimensional Kuramoto model. Compared to Zhang and Zhu's recent work in [38], we do not rely on the consistency of one-step foward Euler scheme but analyze the discrete model directly to obtain sharper and more explicit result. More precisely, we present the optimal condition for the convergence and order preserving for identical oscilators with generic initial data. Then, we give the exact convergence rate of the identical oscillators to their limit under the reasonable assuption on time step. Finally, we provide an alternative proof of the asymptotic phase-locking of nonidentical oscillators which can be applied whenever the given Lyapunov functional is continuous and all zeros are isolated.

Stability of a non-local kinetic model for cell migration with density dependent orientation bias
Nadia Loy and Luigi Preziosi
2020, 13(5): 1007-1027 doi: 10.3934/krm.2020035 +[Abstract](211) +[HTML](46) +[PDF](1760.4KB)
Abstract:

The aim of the article is to study the stability of a non-local kinetic model proposed in [17], that is a kinetic model for cell migration taking into account the non-local sensing performed by a cell in order to decide its direction and speed of movement. We show that pattern formation results from modulation of one non-dimensional parameter that depends on the tumbling frequency, the sensing radius, the mean speed in a given direction, the uniform configuration density and the tactic response to the cell density. Numerical simulations show that our linear stability analysis predicts quite precisely the ranges of parameters determining instability and pattern formation. We also extend the stability analysis to the case of different mean speeds in different directions.

Gelfand-Shilov smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off
Wei-Xi Li and Lvqiao Liu
2020, 13(5): 1029-1046 doi: 10.3934/krm.2020036 +[Abstract](305) +[HTML](46) +[PDF](489.1KB)
Abstract:

In this work we consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation. For any given solution belonging to weighted Sobolev space, we will show it enjoys at positive time the Gelfand-Shilov smoothing effect for the velocity variable and Gevrey regularizing properties for the spatial variable. This improves the result of Lerner-Morimoto-Pravda-Starov-Xu [J. Funct. Anal. 269 (2015) 459-535] on one-dimensional Boltzmann equation to the physical three-dimensional case. Our proof relies on the elementary \begin{document}$ L^2 $\end{document} weighted estimate.

Weak dissipative solutions to a free-boundary problem for finitely extensible bead-spring chain molecules: Variable viscosity coefficients
Donatella Donatelli, Tessa Thorsen and Konstantina Trivisa
2020, 13(5): 1047-1070 doi: 10.3934/krm.2020037 +[Abstract](228) +[HTML](38) +[PDF](483.75KB)
Abstract:

We investigate the global existence of weak solutions to a free boundary problem governing the evolution of finitely extensible bead-spring chains in dilute polymers. The free boundary in the present context is defined with regard to a density threshold of \begin{document}$ \rho = 1, $\end{document} below which the fluid is modeled as compressible and above which the fluid is modeled as incompressible. The present article focuses on the physically relevant case in which the viscosity coefficients present in the system depend on the polymer number density, extending the earlier work [8]. We construct the weak solutions of the free boundary problem by performing the asymptotic limit as the adiabatic exponent \begin{document}$ \gamma $\end{document} goes to \begin{document}$ \infty $\end{document} for the macroscopic model introduced by Feireisl, Lu and Süli in [10] (see also [6]). The weak sequential stability of the family of dissipative (finite energy) weak solutions to the free boundary problem is also established.

2019  Impact Factor: 1.311

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