This is a joint work with Nader Masmoudi (Courant Institute). We show that solutions of the nonlinear Klein-Gordon equation can be approximated by solutions to a system of two nonlinear Schrodinger equations as the speed of light goes to infinity, in the strong topology of the energy. We obtain the boundedness and compactness in the energy space by appropriate separation of frequency in the space-time norms of Strichartz type. Although the energy conservation does not imply any uniform global bound of the H^1 norm because of divergence of the mass energy, we can derive a priori bound of H^1 norm via uniform global space-time estimates for another approximating system, by the method which has been developed recently in the scattering theory.
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