On the Cutoff Approximation for the Boltzmann Equation with Long-Range Interaction

The Boltzmann collision operator for long-range interactions is usually employed in its "weak form" in the literature. However the weak form utilizes the symmetry property of the spherical integral and thus should be understood more or less in the principle value sense especially for strong angular singularity. To study the integral in the Lebesgue sense, it is natural to define the collision operator via the cutoff approximation. In this way, we give a rigorous proof to the local well-posedness of the Boltzmann equation with the long-range interactions. The result has the following main features and innovations: (1). The initial data is not necessarily a small perturbation around equilibrium but satisfies compatible conditions. (2). A quasi-linear method instead of the standard linearization method is used to prove existence and non-negativity of the solution in a suitably designed energy space depending heavily on the initial data. In such space, we derive the first uniqueness result for the equation in particular for hard potential case.