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Well-posedness Theory and Long-time Behaviour for Linear type Boltzmann Equations under Non-eqilibrium Conditions / Miele, Nicola. - (2025 Oct 03).

Well-posedness Theory and Long-time Behaviour for Linear type Boltzmann Equations under Non-eqilibrium Conditions

MIELE, NICOLA
2025-10-03
3-ott-2025
This thesis is concerned with the analysis of two linear kinetic equations that arise in the context of the dynamics of dilute gases: the Boltzmann-Rayleigh equation for homoenergetic flows and the inelastic Boltzmann–Lorentz equation for granular media. We consider both equations under nonequilibrium conditions. Our aim is twofold: to establish well-posedness results for both equations and to study the long-time behaviour of their solutions. In the first part of the work we focus on the homoenergetic solutions for the Boltzmann- Rayleigh equation. These are a special class of non-equilibrium solutions, invariant under a group of affine transformations, which describes the dynamics of a gas due to collision and the action of a mechanical deformation as shear, dilation, or a combination of both. We first prove existence and uniqueness of global-in-time solutions for different possible choices of the collision kernel. We then analyze the different possible long-time asymptotics of the solution, in the case of simple shear deformation, for different choices of the collision kernel. Our analysis highlights the interplay between deformation-induced transport dynamics and the collision dynamics, which occurs within the gas. We further discuss some conjectures and open problems in this context. In the second part, we consider the inelastic Boltzmann–Lorentz equation which models, at mesoscopic level, the evolution of a tagged particle undergoing inelastic collisions with a fixed background of obstacles. The dissipative character of the collisions, caused by the loss of kinetic energy along collisions, strongly affects the dynamics. More precisely, in this work, we consider the inelastic Boltzmann–Lorentz equation in presence of an external acceleration field, specifically gravity, in the case of Maxwell molecules. After providing a well-posedness result, we prove existence, uniqueness, and stability of the stationary non-equilibrium solution.
Well-posedness Theory and Long-time Behaviour for Linear type Boltzmann Equations under Non-eqilibrium Conditions / Miele, Nicola. - (2025 Oct 03).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12571/35745
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