Estimates for the kinetic transport equation in hyperbolic Sobolev spaces

Abstract

We establish smoothing estimates in the framework of hyperbolic Sobolev spaces for the velocity averaging operator ρ of the solution of the kinetic transport equation. If the velocity domain is either the unit sphere or the unit ball, then, for any exponents q and r, we find a characterisation of the exponents β + and β − , except possibly for an endpoint case, for which DHere, D + and D − are the classical and hyperbolic derivative operators, respectively. In fact, we shall provide an argument which unifies these velocity domains and the velocity averaging estimates in either case are shown to be equivalent to mixed-norm bounds on the cone multiplier operator acting on L 2 . We develop our ideas further in several ways, including estimates for initial data lying in certain Besov spaces, for which a key tool in the proof is the sharp ℓ p decoupling theorem recently established by Bourgain and Demeter. We also show that the level of permissible smoothness increases significantly if we restrict attention to initial data which are radially symmetric in the spatial variable.

Smoothing Estimates for the Kinetic Transport Equation at the Critical Regularity

“…We shall see later that there is a natural way to unify the smoothing estimates we seek for these velocity domains, so for the sake of simplicity of the exposition, we focus this introductory discussion on the unit sphere. As proved in the work of In very recent work [1], the purely L 2 -based results in [8] and [7] were significantly extended to estimates of the form…”

“…For d = 2, this is because the endpoint Strichartz estimate occurs at (q, r) = (4, ∞), and for d = 3, this is because of the absence of a Strichartz estimate for the wave propagator in the case (d, q, r) = (3, 2, ∞). The details of these arguments will not be of particular benefit for the current paper, so we simply refer the reader to [1,Section 5].…”

“…Once Theorem 4.2 is obtained, a scaling argument yields Theorem 4.1. To establish the critical case λ = λ * in the region 1 q > d 2 ( 1 2 − 1 r ), we can follow the structure of the argument above in Section 3, and the remaining estimates in non-critical cases can be proved following the argument in [1]; thus, we present only an outline here.…”

“…for general q, r ∈ [2, ∞) and where ρ is either ρ s or ρ b . In this case, the scaling condition β + + β − = d r + 1 q − d 2 is necessary for (1.2), and examples in [1] show that β − ≥ 1 q + d−1 2r − d−1 2 and β − > 1−d 4 are also necessary conditions. Furthermore, it was shown in [1] that (1.2)…”

“…As already noted above, when (q, r) = (2, 2), this can be found in [8] in two spatial dimensions and [8] in general. Also, it was noted in [1] that (1.2) holds in the critical case β − = β * − for general q = r by utilising the cone multiplier estimates in [19]. Such cone multiplier estimates ultimately relied upon the bilinear theory for the Fourier restriction problem.…”