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1
Content available On the one-dimensional mean-field games with congestion
Sedjro Marc
Mathematica Applicanda
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2022
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Vol. 50, no. 1
139--151
EN
In this work, we consider a one dimensional forward-forward model of Mean-field Games with congestion. We establish a connection between such models and conservation laws. Next, we show the existence of a non trivial convex entropies. Finally, we investigate the existence of solutions in the parabolic case and derived some estimates thanks to the existence of such convex entropies.
PL
Praca poświęcona jest jednowymiarowym modelom gier z uśrednionym oddziaływaniem w zarządzaniu. Takie badania mają na celu analizę podejmowania strategicznych decyzji przez czynniki mało oddziaływające w bardzo dużych populacjach. Ustalany jest związek między takimi modelami a prawami zachowania. W wyniku tych badań pokazano istnienie nietrywialnych entropii wypukłych. W końcowej części badane jest istnienie rozwiązań w przypadku parabolicznym i wyprowadzane są pewne oszacowania z istnienia takich wypukłych entropii.
2
Content available Perturbation theory, M-essential spectra of 2 x 2 operator matrices and application to transport operators
Jeribi Aref, Moalla Nedra, Yengui Sonia
Journal of Mathematics and Applications
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2021
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Vol. 44
35--55
EN
In this article we give some results on perturbation theory of 2 x 2 block operator matrices on the product of Banach spaces. Furthermore, we investigate their M-essential spectra. Finally, we apply the obtained results to determine the M-essential spectra of two group transport operators with general boundary conditions in the Banach space Lp([-a, a] x [-1, 1]) x Lp([-a, a] x [-1, 1]), p ≥ 1 and a > 0.
3
Solution of a transport equation with discontinuous coefficients
Das Abhishek, Joseph K. T.
Journal of Applied Analysis
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2021
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Vol. 27, nr 2
219--238
EN
In this article, we study initial and initial-boundary value problems for a non-strictly hyperbolic system whose characteristic speed is not smooth and takes values in {−1, 0, 1}. We construct an explicit formula for the weak solution.We also study the interaction of waves and the large time asymptotic behavior of a solution for the case when the initial data is periodic with zero mean over the period and also for the case when the initial data has compact support.
4
Content available Existence of minimizers for optical flow based optimal control problems under mild regularity assumptions
Jarde Philipp, Ulbrich Michael
Control and Cybernetics
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2019
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Vol. 48, No. 2
261--306
EN
Optimal control problems governed by a transport equation are investigated that are motivated by optical flow problems. The control is given by the velocity field, corresponding to the optical flow, while the state corresponds to the brightness of image points. The problem is studied in the setting of spatially BVregular vector fields under very low regularity requirements. Existing stability results for the control-to-state operator are improved and based on this the existence of minimizers for several classes of optimal control problems is proved under mild assumptions on the admissible sets.
5
Content available Modelling of transient heat transport in metal films using the interval lattice Boltzmann method
Piasecka-Belkhayat A., Korczak A.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2016
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Vol. 64, nr 3
599--606
EN
In the paper a description of heat transfer in one-dimensional crystalline solids is presented. The lattice Boltzmann method based on Boltzmann transport equation is used to simulate the nanoscale heat transport in thin metal films. The coupled lattice Boltzmann equations for electrons and phonons are applied to analyze the heating process of thin metal films via laser pulse. Such approach in which the parameters appearing in the problem analyzed are treated as constant values is widely used, but in the paper the interval values of relaxation times and electron-phonon coupling factor are taken into account. The problem formulated has been solved by means of the interval lattice Boltzmann method using the rules of directed interval arithmetic. In the final part of the paper the results of numerical computations are shown.
6
Content available remote On continuous time version of two-phase cell cycle model of Tyrcha
Zwoleński P.
Mathematica Applicanda
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2013
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Vol. 41, No. 1
13--31
EN
We consider a model of two-phase cell cycle in a maturity-structured cellular population, which consists of a system of first order linear partial differential equations (transport equations). The model is based on similar biological assumptions as models of Lasota-Mackey, Tyson-Hannsgen and Tyrcha. We examine behavior of the solutions of the system along characteristics, give conditions for existence of invariant density, and compare results with outcomes of generation.
PL
W artykule rozważamy matematyczny opis dwufazowego modelu cyklu komórkowego w populacjach ze struktura dojrzałości komórkowej. Model, którym się zajmujemy, składa się z układu dwóch równań rózniczkowych cząstkowych (równań transportu) i jest budowany na podobnych biologicznych założeniach co modele Lasoty-Mackeya, Tysona-Hannsgena oraz Tyrchy. Rozwiązania otrzymanego układu równań badamy wzdłuż charakterystyk, podajemy warunki wystarczające na istnienie gęstości niezmienniczej i porównujemy wyniki z wynikami badań nad modelem Tyrchy.
7
Content available remote Support overlapping Markov semigroups
Tyran-Kamińska M.
Bulletin of the Polish Academy of Sciences. Mathematics
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2003
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Vol. 51, no 4
419-438
EN
New sufficient conditions for asymptotic stability of Markov semigroups are given. These criteria are applied to transport equations.
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