A New Approach to the Creation and Propagation of Exponential Moments in the Boltzmann Equation

Abstract

Abstract. We study the creation and propagation of exponential moments of solutions to the spatially homogeneous d-dimensional Boltzmann equation. In particular, when the collision kernel is of the form |v − v * | β b(cos(θ)) for β ∈ (0, 2] with cos(θ) = |v − v * | −1 (v − v * ) · σ and σ ∈ S d−1 , and assuming the classical cut-off condition b(cos(θ)) integrable in S d−1 , we prove that there exists a > 0 such that moments with weight exp(a min{t, 1}|v| β ) are finite for t > 0, where a only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients.Mathematics Subject Classification (2000): 26D10, 35A23, 76P05, 82C40, 82D10

Kac’s program in kinetic theory

“…This method is, we hope, interesting per se for several reasons: (1) it is fully quantitative, (2) it is highly flexible in terms of the functional spaces used in the proof, (3) it requires a minimal amount of informations on the N -particle systems but more stability informations on the limit PDE (we intentionally presented the assumptions in a way resembling the proof of the convergence of a numerical scheme, which was our "methodological model"), (4) the "differential stability" conditions that are required on the limit PDE seem (to our knowledge) new, at least for the Boltzmann equation.…”

Spatially Homogeneous Boltzmann Equation for Relativistic Particles

“…Note that the exponent for p 0 in part (1) of Theorem 5.2 is exactly the growth rate of the kinetic part of the collision kernel. This optimal result was first established for the classical case in [2] We mention that the creation of moments holds only when b > 0. In the case b = 0, we cannot expect the moments higher than the one satisfied by the initial data to be created.…”

“…The four-momentum p μ (μ = 0, 1, 2, 3) is defined by p μ = (p 0 , p), where p 0 = 1 + |p| 2 and p = (p 1 , p 2 , p 3 ) denote the energy and the momentum of a particle respectively. We use | · | to denote the Euclidean norm: |p| 2 = (p…”

“…Various L p estimates and their applications were considered in [6,38,39,48]. For the propagation and creation of various types of moments estimates, see [2,3,7,8,10,11,19,26,28,49,53,45]. It seems that the spatially homogeneous Boltzmann equation is very well understood by now, but very few works have studied the relativistic case.…”

“…From the differential inequality in Lemma 5.1, the following theorem on the creation and propagation of exponential moments in L 1 can be obtained by using the same arguments as in [2,10,11,28]. b dp < ∞ ∀t > t * .…”