Trace theorems and kinetic equations for non divergence free external forces

Abstract

For a general class of time dependent linear Boltzmann type equations with (i) an external, non divergence free force terma a ∂u/∂ξ (ii) a collision term which can be written as the difference of a gain term involving a general nonnegative "collision frequencyn h(x,ξ,t) and a loss term involving an arbitrary bounded linear operator J, and (iii) a general boundary operator K which is a (strict) contraction, the method of characteristics and perturbation techniques are used to obtain the well-posed- ness of the initial-boundary value problem, provided the divergence b of a is bounded above. The functional setting is L^p, 1<p< + ∞ For time independent data the transport operator is shown to generate a C_o-semigroup on L_p(Σdμ). The results are proven by generalizing a recently established theory of time dependent kinetic equations where the external force is divergence free with respect to velocity. Solutions on spaces of measures are discussed briefly.

AMS(MOS):

• 70G35

Keywords:

• Boltzman equation
• Fokker-Planck equation

¹A large part of the research was conducted as a visiting professer supported by C.N.R.(Consiglio Nazionale per le Ricerche), Gruppo Nazionale per la Fisica Matematica. The author has greatly benefited from the hospitality rendered by the Dipartimento di Matematica dell'Universita di Pavia

¹A large part of the research was conducted as a visiting professer supported by C.N.R.(Consiglio Nazionale per le Ricerche), Gruppo Nazionale per la Fisica Matematica. The author has greatly benefited from the hospitality rendered by the Dipartimento di Matematica dell'Universita di Pavia

Notes

¹A large part of the research was conducted as a visiting professer supported by C.N.R.(Consiglio Nazionale per le Ricerche), Gruppo Nazionale per la Fisica Matematica. The author has greatly benefited from the hospitality rendered by the Dipartimento di Matematica dell'Universita di Pavia