Sur le problème de Cauchy pour l’ équation de Kac homogène sans troncature angulaire

Abstract

The thesis addresses several problems. It shows, in first time the existence and stability of the weak solution. Where the condition on the test function and on the initial datum was weaken and becomes optimal, thanks to a buckle of changes of variables this also allowed us to treat the singularities of the kernel of all kinds. Thus, we prove some properties of Kac’s collision operator. Then we establish the moment’s production of all kinds and even of infinite order in L1 and in L2, knowing that there are no preliminary studies, to except that studying their propagation with heavy assumptions. Then we focus on the study of regularizingn effect in the sense of Sobolev and of Gevrey with a weight of order one, which has as an object to improve the result of Desvillettes [19]. Finally, we generalize these last two results with a weight of infinite order, where the Sobolev smouthing effect is done in a manner somewhat similar to that of [47] without any additional assumption about the moments on the solution and on its initial datum. This is a more accurate result than that of latter Reference. That is, the solution is in Htl tN +2 +k, where N and l can reach +∞, as soon as the initial data is in L1, and even if it is in H−k, where k > 0. In all proofs of these results we use the regularizing method with the combination of the Littlewood-Paley decomposition and the partition of unity with respect to the variable v which optimized the condition on the initial datum.