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


Author: Thomas G. Kurtz
Journal: Trans. Amer. Math. Soc. 186 (1973), 259-272
MSC: Primary 47H15
DOI: https://doi.org/10.1090/S0002-9947-1973-0336482-1
Erratum: Trans. Amer. Math. Soc. 209 (1975), 442.
MathSciNet review: 0336482
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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

$\displaystyle {u_t} = - \alpha {u_x} + {\alpha ^2}({v^2} - {u^2})$

and

$\displaystyle {v_t} = \alpha {v_x} + {\alpha ^2}({u^2} - {v^2}),$

$ \alpha $ a constant, approximate the solutions of

$\displaystyle {u_t} = 1/4({d^2}/d{x^2})\,\log \,u$

as $ \alpha $ goes to infinity.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0336482-1
Keywords: Nonlinear semigroups, approximation, gas kinetics
Article copyright: © Copyright 1973 American Mathematical Society

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