Well-Posedness of Spatially Homogeneous Boltzmann Equation with Full-Range Interaction

Abstract

In the present work, we give the coercivity estimates for the Boltzmann collision operator Q(·, ·) without angular cut-off to clarify in which case the functional −Q(g, f ), f will become the truly sub-elliptic. Based on this observation and commutator estimates in Alexandre et al. (Arch Rat Mech Anal 198:39-123, 2010), the upper bound estimates for the collision operator in Chen and He (Arch Rat Mech Anal 201(2):501-548, 2011) and the stability results in Desvillettes and Mouhot (Arch Rat Mech Anal 193(2):227-253, 2009), in the function space L 1 q ∩ H N , we establish global well-posedness or local well-posedness for the spatially homogeneous Boltzmann equation with full-range interaction(covering most of physical collision kernels).

High order approximation for the Boltzmann equation without angular cutoff

“…. Now patching together the inequalities (43), ( 44) and (45), and using the basic inequality (10), we have…”

“…Recently, Lu-Mouhot in [14] extended the results to the space of non-negative measure with finite non-increasing kinetic energy. For the well-posedness of the spatially homogeneous Boltzmann equation without angular cutoff, we refer to [10] and the references therein. As for the regularity theory of the equation, we refer to [18] for the analysis of the positive part of the collision operator and the propagation of smoothness in the case of angular cutoff and refer to [2], [4], [13] and [19] in the case of long-range interaction.…”

“…Step 1.1 : (Uniform Upper Bound in L 1 l ) In this step, we shall use the energy method to get the uniform upper bound of L 1 l norm of {f n } n with respect to n. Applying the basic inequality (10), for any η > 0, there holds…”

High order approximation for the Boltzmann equation without angular cutoff under moderately soft potentials

“…does not hold any more. As a consequence, we only have time dependent estimates (24) and (25). Roughly speaking, we only get E(f (t)) ≤ C (t, E(f 0 )) in proposition 3 on a priori estimate.…”

“…The techniques in [1] were later used to prove sub-elliptic coercivity estimate which is crucial to the well-posedness and regularity theory. For well-posedness theory, we refer [3] for existence of renormalized solutions, and [11], [17], [25], [18], [15] for existence and uniqueness of classic solutions. Thanks to smoothing effect of the non-cutoff kernel, regularity results are very fruitful, see [2], [9], [28], [46] for instance.…”