Approximation of the Linear Boltzmann Equation by the Fokker-Planck Equation

Unique representation of an inverse-kinetic theory for incompressible Newtonian fluids

“…C) Finally, since thanks to A) {ρ, V,p} are necessarily classical solutions of INSE, it follows that they fulfill necessarily also the energy equation (8). Hence, (44) and (35) coincide identically in Γ × I.…”

“…Hence, the position (32) does not conflict with the Pawula theorem [7,8] which yields a sufficient condition for the positivity of the kinetic distribution function f. The condition of positivity for the kinetic distribution function satisfying the inverse kinetic equation (9) which corresponds to the definition (32) for F 0 (α) has been investigated elsewhere [4]. In particular by assuming that f results initially strictly positive and suitably smooth, one can prove that f satisfies an H-theorem both for Maxwellian and non-Maxwellian distributions functions.…”

“…As a further development, let us now impose the additional requirement that the inverse kinetic theory yields explicitly also the energy equation (8).…”

“…Desirable features of the inverse kinetic theory involve the requirement that, under suitable assumptions, the functional form of the relevant inverse kinetic equation, yielding the INSE equations, be uniquely defined. In addition, it might be convenient to impose also the validity of additional fluid equations, such for example the energy equation (8). In fact, it is well known that the energy equation is not satisfied by weak solutions of INSE and, as a consequence, also by certain numerical schemes.…”

“…This means that arbitrary contributions in the kinetic equation, which vanish identically under a such an hypothesis, can be included in the same kinetic equation. Consistently with the previous regularity assumption, here we intend to consider, in particular, the requirement that the inverse kinetic equation yields also the energy equation (8).…”