Global well-posedness for the Sobolev critical nonlinear Schrödinger system in four space dimensions

Abstract

We consider the Cauchy problem of the 3-component system of nonlinear Schrödinger equations with cubic nonlinearity in four space dimensions. The system is deduced by assuming the gauge invariant mass resonant type and the mass and energy conservation laws hold. We prove the global well-posedness for the system below the ground state under the mass resonance condition. The argument is following the well-known results due to Kenig–Merle [22] and the global solution we obtain here scatters.