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Networks & Heterogeneous Media

March 2018 , Volume 13 , Issue 1

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Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours
Roberto Alicandro, Giuliano Lazzaroni and Mariapia Palombaro
2018, 13(1): 1-26 doi: 10.3934/nhm.2018001 +[Abstract](508) +[HTML](199) +[PDF](360.33KB)
Abstract:

We study continuum limits of discrete models for (possibly heterogeneous) nanowires. The lattice energy includes at least nearest and next-to-nearest neighbour interactions: the latter have the role of penalising changes of orientation. In the heterogeneous case, we obtain an estimate on the minimal energy spent to match different equilibria. This gives insight into the nucleation of dislocations in epitaxially grown heterostructured nanowires.

Stochastic homogenization of maximal monotone relations and applications
Luca Lussardi, Stefano Marini and Marco Veneroni
2018, 13(1): 27-45 doi: 10.3934/nhm.2018002 +[Abstract](345) +[HTML](164) +[PDF](488.35KB)
Abstract:

We study the homogenization of a stationary random maximal monotone operator on a probability space equipped with an ergodic dynamical system. The proof relies on Fitzpatrick's variational formulation of monotone relations, on Visintin's scale integration/disintegration theory and on Tartar-Murat's compensated compactness. We provide applications to systems of PDEs with random coefficients arising in electromagnetism and in nonlinear elasticity.

Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network
Francesca R. Guarguaglini
2018, 13(1): 47-67 doi: 10.3934/nhm.2018003 +[Abstract](463) +[HTML](206) +[PDF](524.21KB)
Abstract:

This paper approaches the question of existence and uniqueness of stationary solutions to a semilinear hyperbolic-parabolic system and the study of the asymptotic behaviour of global solutions. The system is a model for some biological phenomena evolving on a network composed by a finite number of nodes and oriented arcs. The transmission conditions for the unknowns, set at each inner node, are crucial features of the model.

On a vorticity-based formulation for reaction-diffusion-Brinkman systems
Verónica Anaya, Mostafa Bendahmane, David Mora and Ricardo Ruiz Baier
2018, 13(1): 69-94 doi: 10.3934/nhm.2018004 +[Abstract](551) +[HTML](398) +[PDF](6157.76KB)
Abstract:

We are interested in modelling the interaction of bacteria and certain nutrient concentration within a porous medium admitting viscous flow. The governing equations in primal-mixed form consist of an advection-reaction-diffusion system representing the bacteria-chemical mass exchange, coupled to the Brinkman problem written in terms of fluid vorticity, velocity and pressure, and describing the flow patterns driven by an external source depending on the local distribution of the chemical species. A priori stability bounds are derived for the uncoupled problems, and the solvability of the full system is analysed using a fixed-point approach. We introduce a primal-mixed finite element method to numerically solve the model equations, employing a primal scheme with piecewise linear approximation of the reaction-diffusion unknowns, while the discrete flow problem uses a mixed approach based on Raviart-Thomas elements for velocity, Nédélec elements for vorticity, and piecewise constant pressure approximations. In particular, this choice produces exactly divergence-free velocity approximations. We establish existence of discrete solutions and show their convergence to the weak solution of the continuous coupled problem. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme.

On Lennard-Jones systems with finite range interactions and their asymptotic analysis
Mathias Schäffner and Anja Schlömerkemper
2018, 13(1): 95-118 doi: 10.3934/nhm.2018005 +[Abstract](366) +[HTML](176) +[PDF](452.77KB)
Abstract:

The aim of this work is to provide further insight into the qualitative behavior of mechanical systems that are well described by Lennard-Jones type interactions on an atomistic scale. By means of $Γ$-convergence techniques, we study the continuum limit of one-dimensional chains of atoms with finite range interactions of Lennard-Jones type, including the classical Lennard-Jones potentials. So far, explicit formula for the continuum limit were only available for the case of nearest and next-to-nearest neighbour interactions. In this work, we provide an explicit expression for the continuum limit in the case of finite range interactions. The obtained homogenization formula is given by the convexification of a Cauchy-Born energy density.

Furthermore, we study rescaled energies in which bulk and surface contributions scale in the same way. The related discrete-to-continuum limit yields a rigorous derivation of a one-dimensional version of Griffith' fracture energy and thus generalizes earlier derivations for nearest and next-to-nearest neighbors to the case of finite range interactions.

A crucial ingredient to our proofs is a novel decomposition of the energy that allows for refined estimates.

Fisher-KPP equations and applications to a model in medical sciences
Benjamin Contri
2018, 13(1): 119-153 doi: 10.3934/nhm.2018006 +[Abstract](333) +[HTML](166) +[PDF](460.08KB)
Abstract:

This paper is devoted to a class of reaction-diffusion equations with nonlinearities depending on time modeling a cancerous process with chemotherapy. We begin by considering nonlinearities periodic in time. For these functions, we investigate equilibrium states, and we deduce the large time behavior of the solutions, spreading properties and the existence of pulsating fronts. Next, we study nonlinearities asymptotically periodic in time with perturbation. We show that the large time behavior and the spreading properties can still be determined in this case.

Green's function for elliptic systems: Moment bounds
Peter Bella and Arianna Giunti
2018, 13(1): 155-176 doi: 10.3934/nhm.2018007 +[Abstract](415) +[HTML](138) +[PDF](482.84KB)
Abstract:

We study estimates of the Green's function in $\mathbb{R}^d$ with $d ≥ 2$, for the linear second order elliptic equation in divergence form with variable uniformly elliptic coefficients. In the case $d ≥ 3$, we obtain estimates on the Green's function, its gradient, and the second mixed derivatives which scale optimally in space, in terms of the "minimal radius" $r_*$ introduced in [Gloria, Neukamm, and Otto: A regularity theory for random elliptic operators; ArXiv e-prints (2014)]. As an application, our result implies optimal stochastic Gaussian bounds on the Green's function and its derivatives in the realm of homogenization of equations with random coefficient fields with finite range of dependence. In two dimensions, where in general the Green's function does not exist, we construct its gradient and show the corresponding estimates on the gradient and mixed second derivatives. Since we do not use any scalar methods in the argument, the result holds in the case of uniformly elliptic systems as well.

Entropy-preserving coupling conditions for one-dimensional Euler systems at junctions
Jens Lang and Pascal Mindt
2018, 13(1): 177-190 doi: 10.3934/nhm.2018008 +[Abstract](398) +[HTML](173) +[PDF](517.43KB)
Abstract:

This paper is concerned with a set of novel coupling conditions for the 3× 3 one-dimensional Euler system with source terms at a junction of pipes with possibly different cross-sectional areas. Beside conservation of mass, we require the equality of the total enthalpy at the junction and that the specific entropy for pipes with outgoing flow equals the convex combination of all entropies that belong to pipes with incoming flow. Previously used coupling conditions include equality of pressure or dynamic pressure. They are restricted to the special case of a junction having only one pipe with outgoing flow direction. Recently, Reigstad [SIAM J. Appl. Math., 75:679-702,2015] showed that such pressure-based coupling conditions can produce non-physical solutions for isothermal flows through the production of mechanical energy. Our new coupling conditions ensure energy as well as entropy conservation and also apply to junctions connecting an arbitrary number of pipes with flexible flow directions. We prove the existence and uniqueness of solutions to the generalised Riemann problem at a junction in the neighbourhood of constant stationary states which belong to the subsonic region. This provides the basis for the well-posedness of the homogeneous and inhomogeneous Cauchy problems for initial data with sufficiently small total variation.

2017  Impact Factor: 1.187

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