Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics

Let A j, A-, be dissipative sets that generate semigroups of nonlinear contractions T At), T St). Conditions are given on } Which imply the existence of a limiting semigroup Tit). The results include types of convergence besides strong convergence. As an application, it is shown that solutions of the pair of equations "2, 2 2* u = -aux + a. v -u) and 2 2 2 vf = avx + a. (u -u), a a constant, approximate the solutions of ut = y (d2/dx2) log u as o. goes to infinity. 1. Introduction. A general theorem concerning the convergence of sequences of semigroups of linear operators was given in [5. The basis of the proof was the following corollary to the Hille-Yosida theorem. Proposition (1. 1). Let T (t) be a strongly continuous semigroup of linear operators on a Banach space L with infinitesimal operator A. Let M be a closed subspace of L. If (A -A) "1: M - M or all k sufficiently large, then Tit): M-*M. Crandall and Liggett 3 have developed a theory for semigroups of nonlinear operators generated by accretive sets that implies essentially the same result. Consequently, many of the results in 5 can now be carried over to nonlinear semigroups of this type.