On the Rate of Explosion for Infinite Energy Solutions of the Spatially Homogeneous Boltzmann Equation

Abstract

Let μ ₀ be a probability measure on ℝ³ representing an initial velocity distribution for the spatially homogeneous Boltzmann equation for pseudo Maxwellian molecules. As long as the initial energy is finite, the solution μ _t will tend to a Maxwellian limit. We show here that if $\int_{\mathbb{R}^{3}}|v|^{2}\mu_{0}({\mathrm{d}}{v})=\infty$ , then instead, all of the mass “explodes to infinity” at a rate governed by the tail behavior of μ ₀. Specifically, for L0, define

$$\eta_{L}=\int_{|v|\le L}|v|^{2}{\mathrm{d}}\mu_{0}(v).$$

Let B _R denote the centered ball of radius R. Then for every R,

$$\lim_{t\to\infty}\int_{B_{R}}{\mathrm{d}}\mu_{t}(v)=0.$$

The explicit rate is estimated in terms of the rate of divergence of η _L . For example, if η _L ≥Const.L ^s, some s>0, $\int_{B_{R}}{\mathrm{d}}\mu_{t}(v)$ is bounded by a multiple of e ^{−[κ3s/(10+9s)]t}, where κ is the absolute value of the spectral gap in the linearized collision operator. Note that in this case, letting B _t denote the ball of radius e ^rt for any r<κ s/(10+9s), we still have $\lim_{t\to\infty}\int_{B_{t}}{\mathrm{d}}\mu_{t}(v)=0$ .

This result shows in particular that the necessary and sufficient condition for lim  _t→∞ μ _t to exist is that the initial data have finite energy. While the “explosion” of the mass towards infinity in the case of infinite energy may seem to be intuitively clear, there seems not to have been any proof, even without the rate information that our proof provides, apart from an analogous result, due to the authors, concerning the Kac equation. A class of infinite energy eternal solutions of the Boltzmann equation have been studied recently by Bobylev and Cercignani. Our rate information is shown here to provide a limit on the tails of such eternal solutions.