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

作者：Alonso Ricardo; Canizo Jose A; Gamba Irene; Mouhot Clement^*

来源：Communications in Partial Differential Equations, 2013, 38(1): 155-169.

DOI：10.1080/03605302.2012.715707

• Boltzmann equation
• Differential inequality
• Exponential moments
• Polynomial moments
• Povzner%26apos
• s estimates

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 vertical bar v - v *vertical bar beta b(cos (?)) for beta ? (0, 2] with cos (?) = vertical bar v - v *vertical bar-1(v - v *).s and s ? ?? d-1, and assuming the classical cut-off condition b(cos (?)) integrable in ?? d-1, we prove that there exists a %26gt; 0 such that moments with weight exp (amin {t, 1}vertical bar v vertical bar beta) are finite for t %26gt; 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.

• 出版日期2013-1-1