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Elliptic Systems (elliptic + system)

Distribution by Scientific Domains

Mathematics and Statistics	90%

Selected Abstracts

Blow-up estimates for a quasi-linear reaction,diffusion system

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2003
Yang Zuodong
In this paper, some sufficient conditions under which the quasilinear elliptic system -div(,,u,p-2,u) = uv, -div(,,u,q-2,u) = uv in ,N(N,3) has no radially symmetric positive solution is derived. Then by using this non-existence result, blow-up estimates for a class of quasilinear reaction,diffusion systems ut = div (,,u,p-2,u)+uv,vt = div(,,v,q-2,v) +uv with the homogeneous Dirichlet boundary value conditions are obtained. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Eigenvalue asymptotics for a boundary problem involving an elliptic system

MATHEMATISCHE NACHRICHTEN, Issue 11 2006
M. Faierman
Abstract In a recent paper, Agranovich, Denk, and Faierman developed a method for deriving results pertaining to the eigenvalue asymptotics for scalar elliptic boundary problems involving a weight function under limited smoothness assumptions and under an ellipicity with parameter condition. Denk, Faierman, and Möller then used this method to extend the aforementioned results for the scalar case to the case of a homogeneous elliptic systems. However, the method of Agranovich et al. does not carry over to more general elliptic systems of Agmon,Douglis,Nirenberg type. By employing a different method, we are able to overcome this difficulty, and hence in this paper we derive results pertaining to the eigenvalue asymptotics for more general systems of Agmon,Douglis,Nirenberg type and under limited smoothness assumptions. Furthermore, our results not only subsume those of Denk et al., but are derived under much weaker smoothness assumptions. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

2D thermal/isothermal incompressible viscous flows

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2005
Alfredo Nicolás
Abstract 2D thermal and isothermal time-dependent incompressible viscous flows are presented in rectangular domains governed by the Boussinesq approximation and Navier,Stokes equations in the stream function,vorticity formulation. The results are obtained with a simple numerical scheme based on a fixed point iterative process applied to the non-linear elliptic systems that result after a second-order time discretization. The iterative process leads to the solution of uncoupled, well-conditioned, symmetric linear elliptic problems. Thermal and isothermal examples are associated with the unregularized, driven cavity problem and correspond to several aspect ratios of the cavity. Some results are presented as validation examples and others, to the best of our knowledge, are reported for the first time. The parameters involved in the numerical experiments are the Reynolds number Re, the Grashof number Gr and the aspect ratio. All the results shown correspond to steady state flows obtained from the unsteady problem. Copyright © 2005 John Wiley & Sons, Ltd. [source]

On the absence of eigenvalues of Maxwell and Lamé systems in exterior domains

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2004
Sebastian Bauer
Abstract The arguments showing non-existence of eigensolutions to exterior-boundary value problems associated with systems,such as the Maxwell and Lamé system,rely on showing that such solutions would have to have compact support and therefore,by a unique continuation property,cannot be non-trivial. Here we will focus on the first part of the argument. For a class of second order elliptic systems it will be shown that L2 -solutions in exterior domains must have compact support. Both the asymptotically isotropic Maxwell system and the Lamé system with asymptotically decaying perturbations can be reduced to this class of elliptic systems. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Homogenizing the acoustic properties of a porous matrix containing an incompressible inviscid fluid

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2003
J. L. Ferrin
We undertake a rigorous derivation of the Biot's law for a porous elastic solid containing an inviscid fluid. We consider small displacements of a linear elastic solid being itself a connected periodic skeleton containing a pore structure of the characteristic size ,. It is completely saturated by an incompressible inviscid fluid. The model is described by the equations of the linear elasticity coupled with the linearized incompressible Euler system. We study the homogenization limit when the pore size ,tends to zero. The main difficulty is obtaining an a priori estimate for the gradient of the fluid velocity in the pore structure. Under the assumption that the solid part is connected and using results on the first order elliptic systems, we obtain the required estimate. It allows us to apply appropriate results from the 2-scale convergence. Then it is proved that the microscopic displacements and the fluid pressure converge in 2-scales towards a linear hyperbolic system for an effective displacement and an effective pressure field. Using correctors, we also give a strong convergence result. The obtained system is then compared with the Biot's law. It is found that there is a constitutive relation linking the effective pressure with the divergences of the effective fluid and solid displacements. Then we prove that the homogenized model coincides with the Biot's equations but with the added mass ,a being a matrix, which is calculated through an auxiliary problem in the periodic cell for the tortuosity. Furthermore, we get formulas for the matricial coefficients in the Biot's effective stress,strain relations. Finally, we consider the degenerate case when the fluid part is not connected and obtain Biot's model with the relative fluid displacement equal to zero. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Global solution curves for semilinear systems

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2002
Philip Korman
Abstract We study semilinear elliptic systems in two different directions. In the first one we give a simple constructive proof existence of solutions for a class of sublinear systems. Our main results are in the second direction, where we use bifurcation theory to study global solution curves. Crucial to our analysis is proving positivity properties of the corresponding linearized systems. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Some degenerate elliptic systems and applications to cusped plates

MATHEMATISCHE NACHRICHTEN, Issue 4 2007
G. Jaiani
Abstract We study the tension-compression vibration of an elastic cusped plate under (all reasonable) boundary conditions at the cusped edge and given displacements at the non-cusped edge and stresses at the upper and lower faces of the plate. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Eigenvalue asymptotics for a boundary problem involving an elliptic system

On the subdomain-Galerkin/least squares method for 2- and 3-D mixed elliptic problems with reaction terms

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2002
Suh-Yuh Yang
Abstract In this article we apply the subdomain-Galerkin/least squares method, which is first proposed by Chang and Gunzburger for first-order elliptic systems without reaction terms in the plane, to solve second-order non-selfadjoint elliptic problems in two- and three-dimensional bounded domains with triangular or tetrahedral regular triangulations. This method can be viewed as a combination of a direct cell vertex finite volume discretization step and an algebraic least-squares minimization step in which the pressure is approximated by piecewise linear elements and the flux by the lowest order Raviart-Thomas space. This combined approach has the advantages of both finite volume and least-squares methods. Among other things, the combined method is not subject to the Ladyzhenskaya-Babus,ka-Brezzi condition, and the resulting linear system is symmetric and positive definite. An optimal error estimate in the H1(,) × H(div; ,) norm is derived. An equivalent residual-type a posteriori error estimator is also given. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 738,751, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10030. [source]

Optimal boundary control of glass cooling processes

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2004
René Pinnau
Abstract In this paper, an optimal control problem for glass cooling processes is studied. We model glass cooling using the SP1 approximations to the radiative heat transfer equations. The control variable is the temperature at the boundary of the domain. This results in a boundary control problem for a parabolic/elliptic system which is treated by a constrained optimization approach. We consider several cost functionals of tracking-type and formally derive the first-order optimality system. Several numerical methods based on the adjoint variables are investigated. We present results of numerical simulations illustrating the feasibility and performance of the different approaches. Copyright © 2004 John Wiley & Sons, Ltd. [source]