Problèmes mathématiques de l'équation de Boltzmann relativiste

Abstract

This work deals with relativistic Boltzmann equation and more particulary with integral operator of complete equation and integral operator of linearized equation. These operators depend on the differential cross sectionh(〈p, q〉, cos θ) which is a fonction of energy 〈p, q〉 and of the deviation angle θ. The only hypothesis is thath is a symetric function of cosθ. The second part deals essentially with linearized equation in Special Relativity. We take for the distribution function:

$$F\left( {x,p} \right) = a e^{ - \frac{{\lambda p}}{2}} \left( {e^{ - \frac{{\lambda p}}{2}} + \varepsilon f\left( {x,p} \right)} \right)$$

wherea is a constant, λ a constant vector and ɛ a small constant so that ɛ² can be neglected. We obtain the equation:

$$\frac{{p^\alpha }}{{p^0 }}\frac{{\partial f}}{{\partial x^\alpha }} = - K\left( p \right) \cdot f + G\left( f \right)$$

whereK(p) is a positive function andG an Hilbert-Schmidt operator. Then we resolve the Cauchy's problem by taking the Fourier's transformation off, and in the last part by investigating properties of the resolvent of −K+G we establish that asx ⁰→+∞ the solution of this problem has for limit the equilibrium distributiona e ^∓λp.