Mathematical Models and Methods in Applied Sciences

This paper is concerned with large time behavior of solutions to the compressible Navier–Stokes equations in an infinite layer of ${ℝ}^2$ under slip boundary condition. It is shown that if the initial data is sufficiently small, the global solution uniquely exists and the large time behavior of the solution is described by a superposition of one-dimensional diffusion waves.

A family of effective equations that capture the long time dispersive effects of wave propagation in heterogeneous media in an arbitrary large periodic spatial domain $\mathrm{\Omega }\subset {ℝ}^d$ is proposed and analyzed. For a wave equation with highly oscillatory periodic spatial tensors of characteristic length $\epsilon$, we prove that the solution of any member of our family of effective equations is $\epsilon$-close to the true oscillatory wave over a time interval of length $T^{ϵ}={𝒪}({ϵ}^{-2})$ in a norm equivalent to the $L^{\infty }(0,T^{\epsilon };L^2(\mathrm{\Omega }))$ norm. We show that the previously derived effective equation in [T. Dohnal, A. Lamacz and B. Schweizer, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Model. Simulat. 12 (2014) 488–513] belongs to our family of effective equations. Moreover, while Bloch wave techniques were previously used, we show that asymptotic expansion techniques give an alternative way to derive such effective equations. An algorithm to compute the tensors involved in the dispersive equation and allowing for efficient numerical homogenization methods over long time is proposed.

We prove, both for continuous time and discrete time, results establishing convergence to flocking in a model in which each agent is influenced by only a few closest neighbors. We show unconditional convergence to flocking when this number of closest neighbors is at least half of the population and, otherwise, conditional convergence. That is, convergence to flocking provided the initial positions and velocities satisfy an explicit constraint. We also show that the proportion of neighbors necessary for unconditional convergence, one-half, is sharp. Nothing less will do.

This paper deals with the analysis of the asymptotic limit toward the derivation of macroscopic equations for a class of equations modeling complex multicellular systems by methods of the kinetic theory. After having chosen an appropriate scaling of time and space, a Chapman–Enskog expansion is combined with a closed, by minimization, technique to derive hyperbolic models at the macroscopic level. The resulting macroscopic equations show how the macroscopic tissue behavior can be described by hyperbolic systems which seem the most natural in this context. We propose also an asymptotic-preserving well-balanced scheme for the one-dimensional hyperbolic model, in the two-dimensional case, we consider a time-splitting method between the conservative part and the source term where the conservative equation is approximated by the Lax–Friedrichs scheme.

In this paper, we analyse the behaviour of a pile-up of vertically periodic walls of edge dislocations at an obstacle, represented by a locked dislocation wall. Starting from a continuum nonlocal energy $E_{\gamma }$ modelling the interactions — at a typical length-scale of $1/\gamma$ — of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy $E_{\gamma }$ in powers of $1/\gamma$ by $\mathrm{\Gamma }$-convergence, in the limit $\gamma \to \infty$. While the zeroth-order term in the expansion, the $\mathrm{\Gamma }$-limit of $E_{\gamma }$, captures the “bulk” profile of the density of dislocation walls in the pile-up domain, the first-order term in the expansion is a “boundary-layer” energy that captures the profile of the density in the proximity of the lock. This study is a first step towards a rigorous understanding of the behaviour of dislocations at obstacles, defects, and grain boundaries.

We consider the two-dimensional (2D) Landau–de Gennes energy with several elastic constants, subject to general $k$-radially symmetric boundary conditions. We show that for generic elastic constants the critical points consistent with the symmetry of the boundary conditions exist only in the case $k=2$. In this case we identify three types of radial profiles: with two, three of full five components and numerically investigate their minimality and stability depending on suitable parametres. We also numerically study the stability properties of the critical points of the Landau–de Gennes energy and capture the intricate dependence of various qualitative features of these solutions on the elastic constants and the physical regimes of the liquid crystal system.