Moment Inequalities and High-Energy Tails for Boltzmann Equations with Inelastic Interactions

Abstract

We study the high-energy asymptotics of the steady velocity distributions for model systems of granular media in various regimes. The main results obtained are integral estimates of solutions of the hard-sphere Boltzmann equations, which imply that the velocity distribution functions f (v) behave in a certain sense as C exp(−r|v| s ) for |v| large. The values of s, which we call the orders of tails, range from s = 1 to s = 2, depending on the model of external forcing. The method we use is based on the moment inequalities and careful estimating of constants in the integral form of the Povzner-type inequalities.

Lagrangian Numerical Approximations to One‐Dimensional Convolution‐Diffusion Equations

“…It is a well-known fact that for great values of |v|, the solution of (4.1) with γ = 1 behaves like exp(−|v| 3 ), see [7,8,9]. We tried to check it out by iterating (4.4) with σ = 1 for quite a long time starting from a Gaussian initial datum.…”

“…Following [12], we are most of all interested in classical solutions even if we shall keep on using numerical schemes based on the "reciprocal equation" (1.5). Convergence is shown in section 4.1 whereas the "tails" of the solutions for large |v| (see [2,7,8,9,13]) are to be studied numerically in section 4.2. Section 5 is entirely devoted to the study of the Hele-Shaw cell's equation for which we can propose a nonnegativity-preserving scheme based on similar ideas.…”

Spatially Homogeneous Boltzmann Equation for Relativistic Particles

“…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.…”

“…Therefore, the Povzner inequality can be considered as an extension of the conservation laws to general exponents larger than 2. Basically, it says that although we cannot expect exact cancellations for general k > 2, we can still extract a good term which enables us to derive various nice moment estimates [3,7,8,10,11,19,26,28,53,45]. In this paper, we start from establishing the relativistic version of the Povzner inequality below…”

“…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 * .…”