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Networks & Heterogeneous Media

December 2010 , Volume 5 , Issue 4

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Coupling conditions for the $3\times 3$ Euler system
Rinaldo M. Colombo and Francesca Marcellini
2010, 5(4): 675-690 doi: 10.3934/nhm.2010.5.675 +[Abstract](712) +[PDF](477.0KB)
Abstract:
This paper is devoted to the extension to the full $3\times3$ Euler system of the basic analytical properties of the equations governing a fluid flowing in a duct with varying section. First, we consider the Cauchy problem for a pipeline consisting of 2 ducts joined at a junction. Then, this result is extended to more complex pipes. A key assumption in these theorems is the boundedness of the total variation of the pipe's section. We provide explicit examples to show that this bound is necessary.
Classical solutions and feedback stabilization for the gas flow in a sequence of pipes
Markus Dick, Martin Gugat and Günter Leugering
2010, 5(4): 691-709 doi: 10.3934/nhm.2010.5.691 +[Abstract](697) +[PDF](459.4KB)
Abstract:
We consider the subcritical flow in gas networks consisting of a finite linear sequence of pipes coupled by compressor stations. Such networks are important for the transportation of natural gas over large distances to ensure sustained gas supply. We analyse the system dynamics in terms of Riemann invariants and study stationary solutions as well as classical non-stationary solutions for a given finite time interval. Furthermore, we construct feedback laws to stabilize the system locally around a given stationary state. To do so we use a Lyapunov function and prove exponential decay with respect to the $L^2$-norm.
The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift
Patrick Henning and Mario Ohlberger
2010, 5(4): 711-744 doi: 10.3934/nhm.2010.5.711 +[Abstract](672) +[PDF](1615.5KB)
Abstract:
This contribution is concerned with the formulation of a heterogeneous multiscale finite elements method (HMM) for solving linear advection-diffusion problems with rapidly oscillating coefficient functions and a large expected drift. We show that, in the case of periodic coefficient functions, this approach is equivalent to a discretization of the two-scale homogenized equation by means of a Discontinuous Galerkin Time Stepping Method with quadrature. We then derive an optimal order a-priori error estimate for this version of the HMM and finally provide numerical experiments to validate the method.
Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients
Jérôme Coville, Nicolas Dirr and Stephan Luckhaus
2010, 5(4): 745-763 doi: 10.3934/nhm.2010.5.745 +[Abstract](689) +[PDF](336.1KB)
Abstract:
We consider a so-called random obstacle model for the motion of a hypersurface through a field of random obstacles, driven by a constant driving field. The resulting semi-linear parabolic PDE with random coefficients does not admit a global nonnegative stationary solution, which implies that an interface that was flat originally cannot get stationary. The absence of global stationary solutions is shown by proving lower bounds on the growth of stationary solutions on large domains with Dirichlet boundary conditions. Difficulties arise because the random lower order part of the equation cannot be bounded uniformly.
Groundwater flow in a fissurised porous stratum
Michiel Bertsch and Carlo Nitsch
2010, 5(4): 765-782 doi: 10.3934/nhm.2010.5.765 +[Abstract](687) +[PDF](413.5KB)
Abstract:
In [2] Barenblatt e.a. introduced a fluid model for groundwater flow in fissurised porous media. The system consists of two diffusion equations for the groundwater levels in, respectively, the porous bulk and the system of cracks. The equations are coupled by a fluid exchange term. Numerical evidence in [2, 8] suggests that the penetration depth of the fluid increases dramatically due to the presence of cracks and that the smallness of certain parameter values play a key role in this phenomenon. In the present paper we give precise estimates for the penetration depth in terms of the smallness of some of the parameters.
Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel
Grigory Panasenko and Ruxandra Stavre
2010, 5(4): 783-812 doi: 10.3934/nhm.2010.5.783 +[Abstract](626) +[PDF](519.7KB)
Abstract:
The non-steady viscous flow in a thin channel with elastic wall is considered. The viscosity is constant everywhere except for some small neighborhood of the origin of the coordinate system, where it is a variable function. The problem contains two small parameters: $\varepsilon$, that is the ratio of the thickness of the channel and its length, and $ \delta = \varepsilon^\gamma, $ $ \gamma \geq 3 ,$ that is the "softness of the wall", i.e. its inverse (rigidity) is great. An asymptotic expansion of the solution is constructed and, in particular, the leading term is described. An important new element of this paper is the procedure of construction of the boundary layer in the neighborhood of the origin of the coordinate system, generated by the variable viscosity. The error estimates for the difference of a truncated asymptotic ansatz and the exact solution are obtained. To this end, the existence and uniqueness of the solution are studied and some a priori estimates are proved.
Mathematical and numerical analysis for Predator-prey system in a polluted environment
Verónica Anaya, Mostafa Bendahmane and Mauricio Sepúlveda
2010, 5(4): 813-847 doi: 10.3934/nhm.2010.5.813 +[Abstract](715) +[PDF](7625.0KB)
Abstract:
In this paper, we prove existence results for a Predator-prey system in a polluted environment. The existence result is proved by the Schauder fixed-point theorem. Moreover, we construct a combined finite volume - finite element scheme to our model, we establish existence of discrete solutions to this scheme, and show that it converges to a weak solution. The convergence proof is based on deriving series of a priori estimates and using a general $L^p$ compactness criterion. Finally we give some numerical examples.

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