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.