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
HTML articles powered by AMS MathViewer

by Thomas G. Kurtz PDF

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

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.

References

H. Brézis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Functional Analysis 9 (1972), 63–74. MR 0293452, DOI 10.1016/0022-1236(72)90014-6
Howard E. Conner, Some general properties of a class of semilinear hyperbolic systems analogous to the differential-integral equations of gas dynamics, J. Differential Equations 10 (1971), 188–203. MR 289945, DOI 10.1016/0022-0396(71)90046-5
M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265–298. MR 287357, DOI 10.2307/2373376
Thomas G. Kurtz, Extensions of Trotter’s operator semigroup approximation theorems, J. Functional Analysis 3 (1969), 354–375. MR 0242016, DOI 10.1016/0022-1236(69)90031-7
Thomas G. Kurtz, A general theorem on the convergence of operator semigroups, Trans. Amer. Math. Soc. 148 (1970), 23–32. MR 256210, DOI 10.1090/S0002-9947-1970-0256210-5
Thomas G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Functional Analysis 12 (1973), 55–67. MR 0365224, DOI 10.1016/0022-1236(73)90089-x
Joel L. Mermin, An exponential limit formula for nonlinear semigroups, Trans. Amer. Math. Soc. 150 (1970), 469–476. MR 262873, DOI 10.1090/S0002-9947-1970-0262873-0
Isao Miyadera, On the convergence of nonlinear semi-groups, Tohoku Math. J. (2) 21 (1969), 221–236. MR 247525, DOI 10.2748/tmj/1178242993
Isao Miyadera, On the convergence of nonlinear semi-groups. II, J. Math. Soc. Japan 21 (1969), 403–412. MR 247526, DOI 10.2969/jmsj/02130403
Isao Miyadera and Shinnosuke Ôharu, Approximation of semi-groups of nonlinear operators, Tohoku Math. J. (2) 22 (1970), 24–47. MR 262874, DOI 10.2748/tmj/1178242858
Mark Pinsky, Differential equations with a small parameter and the central limit theorem for functions defined on a finite Markov chain, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9 (1968), 101–111. MR 228067, DOI 10.1007/BF01851001
H. F. Trotter, Approximation of semi-groups of operators, Pacific J. Math. 8 (1958), 887–919. MR 103420
S. K. Godunov and U. M. Sultangazin, Discrete models of the Boltzmann kinetic equation, Uspehi Mat. Nauk 26 (1971), no. 3(159), 3–51 (Russian). MR 0293952
Jerome A. Goldstein, Approximation of nonlinear semigroups and evolution equations, J. Math. Soc. Japan 24 (1972), 558–573. MR 306994, DOI 10.2969/jmsj/02440558

Markov processes

33