Analyse spectrale et calcul numérique pour l'équation de Boltzmann

Abstract : In this thesis, we study the solutions of the Boltzmann equation. We are interested in the homogeneous framework in which the solution f(t; x; v) depends only on the time t and the velocity v. We consider singular crosssections (non cuto_ case) in the Maxwellian case. For the study of the Cauchy problem, we consider a uctuation of the solution around the Maxwellian distribution then a decomposition of this uctuation in the spectral base associated to the quantum harmonic oscillator At first, we solve numerically the solutions using symbolic computation methods and spectral decomposition of Hermite functions. We consider regular initial data and initial conditions of distribution type. Next, we prove that there is no longer a global solution in time for a large initial condition that changes sign (which does not contradict the global existence of a weak solution for a positive initial condition - see for example Villani Arch. Rational Mech. Anal 1998).
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Ibrahim Jrad. Analyse spectrale et calcul numérique pour l'équation de Boltzmann. Analyse numérique [math.NA]. Normandie Université, 2018. Français. ⟨NNT : 2018NORMR020⟩. ⟨tel-01830061⟩

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