Stability and Uniqueness for the Spatially Homogeneous Boltzmann Equation with Long-Range Interactions

Abstract

Abstract. In this paper, we prove some a priori stability estimates (in weighted Sobolev spaces) for the spatially homogeneous Boltzmann equation without angular cutoff (covering every physical collision kernels). These estimates are conditioned to some regularity estimates on the solutions, and therefore reduce the stability and uniqueness issue to the one of proving suitable regularity bounds on the solutions. We then prove such regularity bounds for a class of interactions including the so-called (non cutoff and non mollified) hard potentials and moderately soft potentials. In particular, we obtain the first result of global existence and uniqueness for these long-range interactions.

On the Uniqueness for the Spatially Homogeneous Boltzmann Equation with a Strong Angular Singularity

“…Weighted Sobolev spaces were used in [5], while the results of [11] rely on the Kantorovich-Rubinsten distance.…”

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

“…The local and unique solutions were also obtained in the cases of γ + 2s < 0 and s < 1 2 or −3 < γ < 0 and 1 2 ≤ s < 1 respectively. Moreover, in [12], the authors also gave a proposition stating a stability result for strong angular singularity. However, it was only an a priori estimate because of the lack of regularity.…”

“…A few remarks are in order. First, the index N verifying N ≥ 3 is just to make sure that Theorem 1 and Proposition 1 in [12] can be applied to get the uniqueness result. And one may relax it in the case of soft potentials in the spirit of [14].…”

“…Recently, Desvillettes-Mouhot [12], Fournier [13], Fournier-Mouhot [15] and Fournier-Guerin [14] began the study of the stability and uniqueness problems for the Boltzmann equation in the case of true hard potentials and moderate soft potentials with moderate angular singularity or in the case of soft potentials with strong angular singularity (see also [23] for the Maxwellian molecules). In fact, they built the global and unique strong (or weak) solutions in the space of W 1,1 q (or L 1 q ∩ L p ) in the case of γ + 2s > 0 and s < 1 2 .…”

“…Once the regularity is available, the stability results in [12] can be applied to obtain the uniqueness in weighted L 1 space; (2) Revisit the energy method to obtain the uniform estimates for the approximated solutions with respect to by applying the uniform upper and lower bounds for the collision operator established in [9]; (3) Obtain the existence and uniqueness results by using the standard compactness argument and the stability results in [12].…”