Interaction equations of long and short water wave are considered. It is shown that the Cauchy problem for (Equation Presented) is locally well posed in the largest space where the three conservations ∥u(t)∥2 = ∥u0∥∥2, ∥ν(t)∥22 + 2 Im ∫ℝ u(t)∂xū(t) dx = ∥ν0∥22 + 2 Im ∫ℝ u0∂xū0 dx, E(u(t), ν(t)) = E(u0, ν0) can be justified. Here E(u, ν) is the energy functional associated to the system. By these conservation laws, we establish the global well-posedness of the system in the largest class of initial data.
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