Convergence of Sequences of Semigroups of Nonlinear Operators with an Application to Gas Kinetics

Abstract

Let ${A_1},{A_2}, \cdots$ be dissipative sets that generate semigroups of nonlinear contractions ${T_1}(t),{T_2}(t) \cdots$ Conditions are given on $\{ {A_n}\}$ which imply the existence of a limiting semigroup T(t). The results include types of convergence besides strong convergence. As an application, it is shown that solutions of the pair of equations \[ {u_t} = - \alpha {u_x} + {\alpha ^2}({v^2} - {u^2})\] and \[ {v_t} = \alpha {v_x} + {\alpha ^2}({u^2} - {v^2}),\] $\alpha$ a constant, approximate the solutions of \[ {u_t} = 1/4({d^2}/d{x^2}) \log u\] as $\alpha$ goes to infinity.