Existence and uniqueness of solutions to the steady boltzmann equation
Date
1998
Authors
Ghomeshi, Shahin
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Abstract
In this thesis our main objective is to prove the existence and uniqueness of solutions to the Steady Boltzmann Equation with inflow and diffusive boundary conditions, without truncations of the collision kernel. Due to the singularity at v = 0, the non-linear term in the Steady Boltzmann Equation becomes significantly large. In addition to this, another problem with proving existence of steady solutions is that the collision term becomes unbounded as v -> ∞ for hard-sphere interactions.
Previous authors have attacked these problems by imposing unphysical truncations on the collision kernel so that collisions between particles having small velocities are ignored. An additional truncation is then made to control the magnitude of the velocities so that it will not grow without bound.
We present and detail some of the work of Maslova and show that the collision operator has the properties which make it possible to avoid truncations. We introduce several properties of the collision operator which include a series of crucial estimates. These estimates are later used to produce function spaces with the contractive property. General existence and uniqueness is then obtained by an application of the Contraction Mapping Principle.