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Existence, unicité, approximations de solutions d'équations cinétiques et hyperboliques

Abstract : The context of this thesis is particle systems. We deal with different physical systems, described continuously, whose dynamics are modeled by partial differential equations. These equations follow the evolution in time of macroscopic or microscopic quantities, according to scale description. In the first part, we consider a kinetic model for coagulation-fragmentation. We obtain a global existence result, using the notion of DiPerna-Lions renormalized solutions, for initial data satisfying the natural physical bounds, and assumptions of finite L1 and Lp norm (for some p > 1). The second part deals with methods of moments. The aim of these methods is to approximate a kinetic model by a finite number of equations whose unknowns depend only on the space variable. The question is : how to close this system to get a good approximation of the solution of the kinetic model ? In a linear setting, we obtain an explicit method with linear closure relations, which leads to a fast convergence result. In the last part, we work on modeling of traffic jam taking into account the congestion, using a hyperbolic system with constraints, which occurs in the dynamics of a pressureless gas. By suitably modifying this system, we can model "multi-lane" phenomena, like acceleration, and creation of vacuum. An existence and stability result is proved on this new model.
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Submitted on : Wednesday, December 11, 2013 - 10:12:09 AM
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Damien Broizat. Existence, unicité, approximations de solutions d'équations cinétiques et hyperboliques. Mathématiques générales [math.GM]. Université Nice Sophia Antipolis, 2013. Français. ⟨NNT : 2013NICE4052⟩. ⟨tel-00916993⟩

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