Convergence to the Landau equation from the truncated BBGKY hierarchy in the weak-coupling limit

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We show the convergence of solutions to non-Markovian hyperbolic equations to the solution of the nonlinear kinetic Landau equation, keeping the full singularity of the Landau kernel. The hyperbolic equations arise from the truncation of the BBGKY hierarchy for interacting particle systems under the kinetic weak-coupling limit assuming propagation of chaos. The result shows the transition from the microscopic reversible dynamics to the irreversible macroscopic equation at the level of partial differential equations. This resolves the issue arising from [15] , where the singular region has been removed artificially. Since the singularity appears in the Landau equation due to the geometry of particle interactions, it is an intrinsic physical property of the weak-coupling limit which is crucial to the understanding of the transition from particle systems to the Landau equation.