The BGK model with external confining potential: existence, long-time behaviour and time-periodic Maxwellian equilibria
The existence and the long time behavior solution of the Boltzmann equation with the external potential force \({\mathbf F}=\nabla_x\Phi\) are analyzed: \[ \partial_t f+{\mathbf v}\cdot\nabla_x f+\nabla_x \Phi\nabla_v f= J \] The BGK approximation of the collision integral \((J= J_{\text{BGK}})\) and the collision integral \((J= J_B(f))\) with three collision invariants \((\int J(f)I_m d{\mathbf v}= 0,\;I_m= n,\;v,|v^2|)\) are investigated. The Cauchy problem for the Boltzmann equation is studied analytically. Some mathematical aspects of the Boltzmann equation solution including the relaxation problem of rarefied gases are investigated. Strong convergence on \(L^1\) to a Maxwellian equilibrium state are proved if the initial datum \(f(r,v, t=0)\geq 0\) and mass, entropy and total energy are bounded. It is shown that in the case of a quadratic potential \(\Phi({\mathbf x})= \sum_j a_i x_j+{\mathbf b}\cdot{\mathbf x}\), an infinite family of time dependent local Maxwellian solutions exists.
scientific article; zbMATH DE number 5606287