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Kinetic & Related Models

March 2017 , Volume 10 , Issue 1

Special issue dedicated to Prof. Peter Markowich on the occasion of his 60th Birthday

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Preface
Anton Arnold, Paola Pietra and Giuseppe Toscani
2017, 10(1): i-ii doi: 10.3934/krm.201701i +[Abstract](202) +[HTML](135) +[PDF](1249.2KB)
Abstract:
Opinion dynamics over complex networks: Kinetic modelling and numerical methods
Giacomo Albi, Lorenzo Pareschi and Mattia Zanella
2017, 10(1): 1-32 doi: 10.3934/krm.2017001 +[Abstract](598) +[HTML](14) +[PDF](4730.8KB)
Abstract:

In this paper we consider the modeling of opinion dynamics over time dependent large scale networks. A kinetic description of the agents' distribution over the evolving network is considered which combines an opinion update based on binary interactions between agents with a dynamic creation and removal process of new connections. The number of connections of each agent influences the spreading of opinions in the network but also the way connections are created is influenced by the agents' opinion. The evolution of the network of connections is studied by showing that its asymptotic behavior is consistent both with Poisson distributions and truncated power-laws. In order to study the large time behavior of the opinion dynamics a mean field description is derived which allows to compute exact stationary solutions in some simplified situations. Numerical methods which are capable to describe correctly the large time behavior of the system are also introduced and discussed. Finally, several numerical examples showing the influence of the agents' number of connections in the opinion dynamics are reported.

Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities
Matteo Bonforte, Jean Dolbeault, Matteo Muratori and Bruno Nazaret
2017, 10(1): 33-59 doi: 10.3934/krm.2017002 +[Abstract](503) +[HTML](8) +[PDF](1219.8KB)
Abstract:

In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy -entropy production inequality which governs the intermediate asymptotics is indeed equivalent to (CKN), and the self-similar profiles are optimal for (CKN).

We establish an explicit symmetry breaking condition by proving the linear instability of the radial optimal functions for (CKN). Symmetry breaking in (CKN) also has consequences on entropy -entropy production inequalities and on the intermediate asymptotics for (WFD). Even when no symmetry holds in (CKN), asymptotic rates of convergence of the solutions to (WFD) are determined by a weighted Hardy-Poincaré inequality which is interpretedas a linearized entropy -entropy production inequality. All our results rely on the study of the bottom of the spectrum of the linearized diffusion operator around the self-similar profiles, which is equivalent to the linearization of (CKN) around the radial optimal functions, and on variational methods.Consequences for the (WFD) flow will be studied in Part Ⅱ of this work.

Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods
Matteo Bonforte, Jean Dolbeault, Matteo Muratori and Bruno Nazaret
2017, 10(1): 61-91 doi: 10.3934/krm.2017003 +[Abstract](364) +[HTML](2) +[PDF](891.2KB)
Abstract:

This paper is the second part of the study. In Part Ⅰ, self-similar solutions of a weighted fast diffusion equation (WFD) were related to optimal functions in a family of subcritical Caffarelli-Kohn-Nirenberg inequalities (CKN) applied to radially symmetric functions. For these inequalities, the linear instability (symmetry breaking) of the optimal radial solutions relies on the spectral properties of the linearized evolution operator. Symmetry breaking in (CKN) was also related to large-time asymptotics of (WFD), at formal level. A first purposPart Ⅱis to give a rigorous justification of this point, that is, to determine the asymptotic rates of convergence of the solutions to (WFD) in the symmetry range of (CKN) as well as in the symmetry breaking range, and even in regimes beyond the supercritical exponent in (CKN). Global rates of convergence with respect to a free energy (or entropy) functional are also investigated, as well as uniform convergence to self-similar solutions in the strong sense of the relative error. Differences with large-time asymptotics of fast diffusion equations without weights are emphasized.

A kinetic games framework for insurance plans
Daniel Brinkman and Christian Ringhofer
2017, 10(1): 93-116 doi: 10.3934/krm.2017004 +[Abstract](276) +[HTML](1) +[PDF](818.6KB)
Abstract:

The dynamics of insurance plans have been under the microscope in recent years due to the controversy surrounding the implementation of the Affordable Care Act (Obamacare) in the United States. In this paper, we introduce a game between an insurance company and an ensemble of customers choosing between several insurance plans. We then derive a kinetic model for the strategies of the insurer and the decisions of the customers and establish the conditions for which a Nash equilibrium exists for some specific customer distributions. Finally, we give some agent-based numerical results for how the plan enrollment evolves over time which show qualitative agreement to "experimental" results in the literature from two plans in the state of Massachusetts.

Balanced growth path solutions of a Boltzmann mean field game model for knowledge growth
Martin Burger, Alexander Lorz and Marie-Therese Wolfram
2017, 10(1): 117-140 doi: 10.3934/krm.2017005 +[Abstract](361) +[HTML](1) +[PDF](834.2KB)
Abstract:

In this paper we study balanced growth path solutions of a Boltzmann mean field game model proposed by Lucas and Moll [15] to model knowledge growth in an economy.Agents can either increase their knowledge level by exchanging ideas in learning events or by producing goods with the knowledge they already have.The existence of balanced growth path solutions implies exponential growth of the overall production in time. We prove existence of balanced growth path solutions if the initial distribution of individuals with respect to their knowledge level satisfiesa Pareto-tail condition. Furthermore we give first insights into the existence of such solutions if in addition to production and knowledge exchange theknowledge level evolves by geometric Brownian motion.

A logistic equation with nonlocal interactions
Luis Caffarelli, Serena Dipierro and Enrico Valdinoci
2017, 10(1): 141-170 doi: 10.3934/krm.2017006 +[Abstract](414) +[HTML](8) +[PDF](517.1KB)
Abstract:

We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms.

More precisely, for populations that propagate according to a Lévy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case.

As ambient space, we can consider:
  $ \bullet $bounded domains,
  $ \bullet $periodic environments,
  $ \bullet $transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one.

In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.

Explicit equilibrium solutions for the aggregation equation with power-law potentials
José A. Carrillo and Yanghong Huang
2017, 10(1): 171-192 doi: 10.3934/krm.2017007 +[Abstract](498) +[HTML](4) +[PDF](511.8KB)
Abstract:

Despite their wide presence in various models in the study of collective behaviors, explicit swarming patterns are difficult to obtain. In this paper, special stationary solutions of the aggregation equation with power-law kernelsare constructed by inverting Fredholm integral operators or byemploying certain integral identities. These solutions are expected tobe the global energy stable equilibria and to characterize the generic behaviorsof stationary solutions for more general interactions.

Self-organized hydrodynamics with density-dependent velocity
Pierre Degond, Silke Henkes and Hui Yu
2017, 10(1): 193-213 doi: 10.3934/krm.2017008 +[Abstract](1081) +[HTML](148) +[PDF](651.5KB)
Abstract:

Motivated by recent experimental and computational results that show a motility-induced clustering transition in self-propelled particle systems, we study an individual model and its corresponding Self-Organized Hydrodynamic model for collective behaviour that incorporates a density-dependent velocity, as well as inter-particle alignment. The modal analysis of the hydrodynamic model elucidates the relationship between the stability of the equilibria and the changing velocity, and the formation of clusters. We find, in agreement with earlier results for non-aligning particles, that the key criterion for stability is \begin{document} $(ρ v(ρ))'≥q 0$ \end{document}, i.e. a nondecreasing mass flux \begin{document} $ρ v(ρ)$ \end{document} with respect to the density. Numerical simulation for both the individual and hydrodynamic models with a velocity function inspired by experiment demonstrates the validity of the theoretical results.

Deterministic particle approximation of the Hughes model in one space dimension
Marco Di Francesco, Simone Fagioli, Massimiliano Daniele Rosini and Giovanni Russo
2017, 10(1): 215-237 doi: 10.3934/krm.2017009 +[Abstract](432) +[HTML](2) +[PDF](1637.2KB)
Abstract:

In this paper we present a new approach to the solution to a generalized version of Hughes' models for pedestrian movements based on a follow-the-leader many particle approximation. In particular, we provide a rigorous global existence result under a smallness assumption on the initial data ensuring that the trace of the solution along the turning curve is zero for all positive times. We also focus briefly on the approximation procedure for symmetric data and Riemann type data. Two different numerical approaches are adopted for the simulation of the model, namely the proposed particle method and a Godunov type scheme. Several numerical tests are presented, which are in agreement with the theoretical prediction.

A kinetic equation for economic value estimation with irrationality and herding
Bertram Düring, Ansgar Jüngel and Lara Trussardi
2017, 10(1): 239-261 doi: 10.3934/krm.2017010 +[Abstract](390) +[HTML](2) +[PDF](744.8KB)
Abstract:

A kinetic inhomogeneous Boltzmann-type equation is proposed to model the dynamics of the number of agents in a large market depending on the estimated value of an asset and the rationality of the agents. The interaction rules take into account the interplay of the agents with sources of public information, herding phenomena, and irrationality of the individuals. In the formal grazing collision limit, a nonlinear nonlocal Fokker-Planck equation with anisotropic (or incomplete) diffusion is derived. The existence of global-in-time weak solutions to the Fokker-Planck initial-boundary-value problem is proved. Numerical experiments for the Boltzmann equation highlight the importance of the reliability of public information in the formation of bubbles and crashes. The use of Bollinger bands in the simulations shows how herding may lead to strong trends with low volatility of the asset prices, but eventually also to abrupt corrections.

On the classical limit of a time-dependent self-consistent field system: Analysis and computation
Shi Jin, Christof Sparber and Zhennan Zhou
2017, 10(1): 263-298 doi: 10.3934/krm.2017011 +[Abstract](444) +[HTML](6) +[PDF](976.0KB)
Abstract:

We consider a coupled system of Schrödinger equations, arising in quantum mechanics via the so-called time-dependent self-consistent field method. Using Wigner transformation techniques we study the corresponding classical limit dynamics in two cases. In the first case, the classical limit is only taken in one of the two equations, leading to a mixed quantum-classical model which is closely connected to the well-known Ehrenfest method in molecular dynamics. In the second case, the classical limit of the full system is rigorously established, resulting in a system of coupled Vlasov-type equations. In the second part of our work, we provide a numerical study of the coupled semi-classically scaled Schrödinger equations and of the mixed quantum-classical model obtained via Ehrenfest's method. A second order (in time) method is introduced for each case. We show that the proposed methods allow time steps independent of the semi-classical parameter(s) while still capturing the correct behavior of physical observables. It also becomes clear that the order of accuracy of our methods can be improved in a straightforward way.

On interfaces between cell populations with different mobilities
Tommaso Lorenzi, Alexander Lorz and Benoît Perthame
2017, 10(1): 299-311 doi: 10.3934/krm.2017012 +[Abstract](374) +[HTML](1) +[PDF](946.0KB)
Abstract:

Partial differential equations describing the dynamics of cell population densities from a fluid mechanical perspective can model the growth of avascular tumours. In this framework, we consider a system of equations that describes the interaction between a population of dividing cells and a population of non-dividing cells. The two cell populations are characterised by different mobilities. We present the results of numerical simulations displaying two-dimensional spherical waves with sharp interfaces between dividing and non-dividing cells. Furthermore, we numerically observe how different ratios between the mobilities change the morphology of the interfaces, and lead to the emergence of finger-like patterns of invasion above a threshold. Motivated by these simulations, we study the existence of one-dimensional travelling wave solutions.

Aggregated steady states of a kinetic model for chemotaxis
Anne Nouri and Christian Schmeiser
2017, 10(1): 313-327 doi: 10.3934/krm.2017013 +[Abstract](307) +[HTML](4) +[PDF](407.3KB)
Abstract:

A kinetic chemotaxis model with attractive interaction by quasistationary chemical signalling is considered. The special choice of the turning operator, with velocity jumps biased towards the chemical concentration gradient, permits closed ODE systems for moments of the distribution function of arbitrary order. The system for second order moments exhibits a critical mass phenomeneon. The main result is existence of an aggregated steady state for supercritical mass.

2016  Impact Factor: 1.261

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