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

Kurtz, Thomas G.

doi:10.1090/S0002-9947-1973-0336482-1

Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics
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by Thomas G. Kurtz

Trans. Amer. Math. Soc. 186 (1973), 259-272

DOI: https://doi.org/10.1090/S0002-9947-1973-0336482-1

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Erratum: Trans. Amer. Math. Soc. 209 (1975), 442.

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.

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Markov processes

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