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DCDS-B

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.

DCDS

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 $O(1)$.
NHM

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.
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