scholarly journals Boundedness of the Stationary Solution to the Boltzmann Equation with Spatial Smearing, Diffusive Boundary Conditions, and Lions' Collision Kernel

2018 ◽  
Vol 50 (6) ◽  
pp. 5761-5782
Author(s):  
Jörg-Uwe Löbus
Keyword(s):  
Boltzmann Equation ◽  
Boundary Conditions ◽  
Stationary Solution ◽  
Collision Kernel ◽  
The Boltzmann Equation ◽  
Diffusive Boundary

Knudsen type group for time in ℝ and related Boltzmann type equations

2021 ◽  
pp. 2150072
Author(s):  
Jörg-Uwe Löbus
Keyword(s):  
Boundary Conditions ◽  
Global Solution ◽  
Physical Space ◽  
Type Equation ◽  
Collision Kernel ◽  
Unique Global Solution ◽  
Initial Value ◽  
Type Group ◽  
Diffusive Boundary ◽  
Diffusive Part

We consider certain Boltzmann type equations on a bounded physical and a bounded velocity space under the presence of both reflective as well as diffusive boundary conditions. We introduce conditions on the shape of the physical space and on the relation between the reflective and the diffusive part in the boundary conditions such that the associated Knudsen type semigroup can be extended to time [Formula: see text]. Furthermore, we provide conditions under which there exists a unique global solution to a Boltzmann type equation for time [Formula: see text] or for time [Formula: see text] for some [Formula: see text] which is independent of the initial value at time 0. Depending on the collision kernel, [Formula: see text] can be arbitrarily small.

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