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Control of the continuity equation with a non local flow

Rinaldo M. Colombo, Michael Herty, Magali Mercier (2011)

ESAIM: Control, Optimisation and Calculus of Variations

This paper focuses on the analytical properties of the solutions to the continuity equation with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posedness of weak entropy solutions in a class of equations comprising these models. Then, under further regularity conditions, we prove the differentiability of solutions with respect to the initial datum and characterize this derivative. A necessary ...

Contrôle par les coefficients dans le modèle Elrod-Adams

Mohamed El Alaoui Talibi, Abdellah El Kacimi (2001)

ESAIM: Control, Optimisation and Calculus of Variations

Dans ce papier, nous étudions un problème de contrôle par les coefficients issu de la lubrification élastohydrodynamique. La variable de contrôle est l’épaisseur du fluide. Le phénomène de cavitation est pris en compte par le modèle Elrod-Adams, connu pour ses performances dans la conservation des débits d’entrée et de sortie. L’idée est de régulariser dans l’équation d’état le graphe d’Heaviside, en l’approchant par une suite de fonctions monotones et régulières. Nous dérivons les conditions d’optimalité...

Contrôle par les coefficients dans le modèle elrod-adams

Mohamed El Alaoui Talibi, Abdellah El Kacimi (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The purpose of this paper is to study a control by coefficients problem issued from the elastohydrodynamic lubrication. The control variable is the film thickness.The cavitation phenomenon takes place and described by the Elrod-Adams model, suggested in preference to the classical variational inequality due to its ability to describe input and output flow. The idea is to use the penalization in the state equation  by approximating the Heaviside graph whith a sequence of monotone and regular functions....

Convergence estimate for second order Cauchy problems with a small parameter

Branko Najman (1998)

Czechoslovak Mathematical Journal

We consider the second order initial value problem in a Hilbert space, which is a singular perturbation of a first order initial value problem. The difference of the solution and its singular limit is estimated in terms of the small parameter ε . The coefficients are commuting self-adjoint operators and the estimates hold also for the semilinear problem.

Convergence of the matrix transformation method for the finite difference approximation of fractional order diffusion problems

Béla J. Szekeres, Ferenc Izsák (2017)

Applications of Mathematics

Numerical solution of fractional order diffusion problems with homogeneous Dirichlet boundary conditions is investigated on a square domain. An appropriate extension is applied to have a well-posed problem on ℝ 2 and the solution on the square is regarded as a localization. For the numerical approximation a finite difference method is applied combined with the matrix transformation method. Here the discrete fractional Laplacian is approximated with a matrix power instead of computing the complicated...

Convergence results for unbounded solutions of first order non-linear differential-functional equations

Henryk Leszczyński (1996)

Annales Polonici Mathematici

We consider the Cauchy problem in an unbounded region for equations of the type either D t z ( t , x ) = f ( t , x , z ( t , x ) , z ( t , x ) , D x z ( t , x ) ) or D t z ( t , x ) = f ( t , x , z ( t , x ) , z , D x z ( t , x ) ) . We prove convergence of their difference analogues by means of recurrence inequalities in some wide classes of unbounded functions.

Convexity and uniqueness in a free boundary problem arising in combustion theory.

Arshak Petrosyan (2001)

Revista Matemática Iberoamericana

We consider solutions to a free boundary problem for the heat equation, describing the propagation of flames. Suppose there is a bounded domain Ω ⊂ QT = Rn x (0,T) for some T > 0 and a function u > 0 in Ω such thatut = Δu,    in Ω,u = 0 and |∇u| = 1,   on Γ := ∂Ω ∩ QT,u(·,0) = u0,     on Ω0,where Ω0 is a given domain in Rn and u0 is a positive and continuous function in Ω0, vanishing on ∂Ω0. If Ω0 is convex and u0 is concave in Ω0, then we show that (u,Ω) is unique and the time sections...

Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential

Guillaume Bal, Wenjia Jing (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This paper analyzes the random fluctuations obtained by a heterogeneous multi-scale first-order finite element method applied to solve elliptic equations with a random potential. Several multi-scale numerical algorithms have been shown to correctly capture the homogenized limit of solutions of elliptic equations with coefficients modeled as stationary and ergodic random fields. Because theoretical results are available in the continuum setting for such equations, we consider here the case of a second-order...

Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary

Oana Ivanovici (2012)

Journal of the European Mathematical Society

In this paper we consider a smooth and bounded domain Ω ⊂ ℝ d of dimension d ≥ 2 with boundary and we construct sequences of solutions to the wave equation with Dirichlet boundary condition which contradict the Strichartz estimates of the free space, providing losses of derivatives at least for a subset of the usual range of indices. This is due to microlocal phenomena such as caustics generated in arbitrarily small time near the boundary. Moreover, the result holds for microlocally strictly convex domains...

Crack detection using electrostatic measurements

Martin Brühl, Martin Hanke, Michael Pidcock (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper we extend recent work on the detection of inclusions using electrostatic measurements to the problem of crack detection in a two-dimensional object. As in the inclusion case our method is based on a factorization of the difference between two Neumann-Dirichlet operators. The factorization possible in the case of cracks is much simpler than that for inclusions and the analysis is greatly simplified. However, the directional information carried by the crack makes the practical implementation...

Crack detection using electrostatic measurements

Martin Brühl, Martin Hanke, Michael Pidcock (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we extend recent work on the detection of inclusions using electrostatic measurements to the problem of crack detection in a two-dimensional object. As in the inclusion case our method is based on a factorization of the difference between two Neumann-Dirichlet operators. The factorization possible in the case of cracks is much simpler than that for inclusions and the analysis is greatly simplified. However, the directional information carried by the crack makes the practical...

Da Prato-Zabczyk's maximal inequality revisited. I.

Jan Seidler (1993)

Mathematica Bohemica

Existence, uniqueness and regularity of mild solutions to semilinear nonautonomous stochastic parabolic equations with locally lipschitzian nonlinear terms is investigated. The adopted approach is based on the factorization method due to Da Prato, Kwapień and Zabczyk.

Currently displaying 381 – 400 of 1901