# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

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## Networks & Heterogeneous Media

March 2007 , Volume 2 , Issue 1

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2007, 2(1): 1-36 doi: 10.3934/nhm.2007.2.1 +[Abstract](2368) +[PDF](391.8KB)
Abstract:
In this paper some new tools for the study of evolution problems in the framework of Young measures are introduced. A suitable notion of time-dependent system of generalized Young measures is defined, which allows us to extend the classical notions of total variation and absolute continuity with respect to time, as well as the notion of time derivative. The main results are a Helly type theorem for sequences of systems of generalized Young measures and a theorem about the existence of the time derivative for systems with bounded variation with respect to time.
2007, 2(1): 37-54 doi: 10.3934/nhm.2007.2.37 +[Abstract](2037) +[PDF](211.8KB)
Abstract:
In this paper, we develop a multi layer method to solve a generalized case of a phytoplankton model introduced in [7]. It is treated by means of a sequence of approximations: the mixed layer is subdivided into a finite number of thin layers within each of which horizontal velocity can be considered constant with respect to depth. Existence, uniqueness and non negativity of solutions are investigated.
2007, 2(1): 55-79 doi: 10.3934/nhm.2007.2.55 +[Abstract](2250) +[PDF](302.5KB)
Abstract:
We consider a diffusion problem on a network on whose nodes we impose Dirichlet and generalized, non-local Kirchhoff-type conditions. We prove well-posedness of the associated initial value problem, and we exploit the theory of sub-Markovian and ultracontractive semigroups in order to obtain upper Gaussian estimates for the integral kernel. We conclude that the same diffusion problem is governed by an analytic semigroup acting on all $L^p$-type spaces as well as on suitable spaces of continuous functions. Stability and spectral issues are also discussed. As an application we discuss a system of semilinear equations on a network related to potential transmission problems arising in neurobiology.
2007, 2(1): 81-97 doi: 10.3934/nhm.2007.2.81 +[Abstract](2528) +[PDF](1489.0KB)
Abstract:
We consider gas flow in pipeline networks governed by the isothermal Euler equations and introduce a new modeling of compressors in gas networks. Compressor units are modeled as pipe–to–pipe intersections with additional algebraic coupling conditions for the compressor behavior. We prove existence and uniqueness of solutions with respect to these conditions and use the results for numerical simulation and optimization of gas networks.
2007, 2(1): 99-125 doi: 10.3934/nhm.2007.2.99 +[Abstract](2154) +[PDF](328.6KB)
Abstract:
Starting from the three-dimensional Newtonian and incompressible Navier-Stokes equations in a compliant straight vessel, we derive a reduced one-dimensional model by an averaging procedure which takes into consideration the elastic properties of the wall structure. In particular, we neglect terms of the first order with respect to the ratio between the vessel radius and length. Furthermore, we consider that the viscous effects are negligible with respect to the propagative phenomena. The result is a one-dimensional nonlinear hyperbolic system of two equations in one space dimension, which describes the mean longitudinal velocity of the flow and the radial wall displacement. The modelling technique here applied to straight cylindrical vessels may be generalized to account for curvature and torsion. An analysis of well posedness is presented which demonstrates, under reasonable hypotheses, the global in time existence of regular solutions.
2007, 2(1): 127-157 doi: 10.3934/nhm.2007.2.127 +[Abstract](2549) +[PDF](330.0KB)
Abstract:
We consider a scalar conservation law with a discontinuous flux function. The fluxes are non-convex, have multiple points of extrema and can have arbitrary intersections. We propose an entropy formulation based on interface connections and associated jump conditions at the interface. We show that the entropy solutions with respect to each choice of interface connection exist and form a contractive semi-group in $L^1$. Existence is shown by proving convergence of a Godunov type scheme by a suitable modification of the singular mapping approach. This extends the results of [3] to the general case of non-convex flux geometries.
2007, 2(1): 159-179 doi: 10.3934/nhm.2007.2.159 +[Abstract](3190) +[PDF](267.2KB)
Abstract:
We consider a hyperbolic conservation law with discontinuous flux. Such a partial differential equation arises in different applications, in particular we are motivated by a model of traffic flow. We provide a new formulation in terms of Riemann Solvers. Moreover, we determine the class of Riemann Solvers which provide existence and uniqueness of the corresponding weak entropic solutions.
2007, 2(1): 181-192 doi: 10.3934/nhm.2007.2.181 +[Abstract](2073) +[PDF](171.3KB)
Abstract:
Multiscale stochastic homogenization is studied for divergence structure parabolic problems. More specifically we consider the asymptotic behaviour of a sequence of realizations of the form
$\frac{\partial u^\omega_\varepsilon}{\partial t}-$div$(a(T_1(\frac{x}{\varepsilon_1})\omega_1, T_2(\frac{x}{\varepsilon_2})\omega_2 ,t, D u^\omega_\varepsilon))=f.$
It is shown, under certain structure assumptions on the random map $a(\omega_1,\omega_2,t,\xi)$, that the sequence $\{u^\omega_\e}$ of solutions converges weakly in $L^p(0,T;W^{1,p}_0(\Omega))$ to the solution $u$ of the homogenized problem $\frac{\partial u}{\partial t} -$div$( b( t,D u )) = f$.

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