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

Abstract

Let A j, A-, • • • be dissipative sets that generate semigroups of nonlinear contractions T At), T St) • • «. Conditions are given on \A } 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 = yÁ(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.

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