Lagrangian averaging for the 1D compressible Euler equations
Harish S. Bhat Razvan C. Fetecau
Discrete & Continuous Dynamical Systems - B 2006, 6(5): 979-1000 doi: 10.3934/dcdsb.2006.6.979
We consider a $1$-dimensional Lagrangian averaged model for an inviscid compressible fluid. As previously introduced in the literature, such equations are designed to model the effect of fluctuations upon the mean flow in compressible fluids. This paper presents a traveling wave analysis and a numerical study for such a model. The discussion is centered around two issues. One relates to the intriguing wave motions supported by this model. The other is the appropriateness of using Lagrangian-averaged models for compressible flow to approximate shock wave solutions of the compressible Euler equations.
keywords: Lagrangian averaging compressible fluids.
A note on a non-local Kuramoto-Sivashinsky equation
Jared C. Bronski Razvan C. Fetecau Thomas N. Gambill
Discrete & Continuous Dynamical Systems - A 2007, 18(4): 701-707 doi: 10.3934/dcds.2007.18.701
In this note we outline some improvements to a result of Hilhorst, Peletier, Rotariu and Sivashinsky [5] on the $L_2$ boundedness of solutions to a non-local variant of the Kuramoto-Sivashinsky equation with additional stabilizing and destabilizing terms. We are able to make the following improvements: in the case of odd data we reduce the exponent in the estimate lim sup$_t\rightarrow \infty$ ||$u$ || $\le C L^{\nu}$ from $\nu = \frac{11}{5}$ to $\nu=\frac{3}{2}$, and for the case of general initial data we establish an estimate of the above form with $\nu = \frac{13}{6}$. We also remove the restrictions on the magnitudes of the parameters in the model and track the dependence of our estimates on these parameters, assuming they are at least $O(1)$.
keywords: global attractors. Kuramoto-Sivashinsky equation
Modelling heterogeneity and an open-mindedness social norm in opinion dynamics
Clinton Innes Razvan C. Fetecau Ralf W. Wittenberg
Networks & Heterogeneous Media 2017, 12(1): 59-92 doi: 10.3934/nhm.2017003

We study heterogeneous interactions in a time-continuous bounded confidence model for opinion formation. The key new modelling aspects are to distinguish between open-minded and closed-minded behaviour and to include an open-mindedness social norm. The investigations focus on the equilibria supported by the proposed new model; particular attention is given to a novel class of equilibria consisting of multiple connected opinion clusters, which does not occur in the absence of heterogeneity. Various rigorous stability results concerning these equilibria are established. We also incorporate the effect of media in the model and study its implications for opinion formation.

keywords: Opinion dynamics bounded confidence heterogeneous societies group connectivity opinion equilibria
Swarming in domains with boundaries: Approximation and regularization by nonlinear diffusion
Razvan C. Fetecau Mitchell Kovacic Ihsan Topaloglu
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-28 doi: 10.3934/dcdsb.2018238

We consider an aggregation model with nonlinear diffusion in domains with boundaries and investigate the zero diffusion limit of its solutions. We establish the convergence of weak solutions for fixed times, as well as the convergence of energy minimizers in this limit. Numerical simulations that support the analytical results are presented. A second key scope of the numerical studies is to demonstrate that adding small nonlinear diffusion rectifies a flaw of the plain aggregation model in domains with boundaries, which is to evolve into unstable equilibria (non-minimizers of the energy).

keywords: Swarm equilibria energy minimizers gradient flow attractors nonlinear diffusion nonsmooth dynamics

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