A New Approach to the Creation and Propagation of Exponential Moments in the Boltzmann Equation |
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Authors: | Ricardo Alonso José A Cañizo Irene Gamba |
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Institution: | 1. Institute for Pure and Applied Mathematics (IPAM) UCLA – (CAAM) , Rice University , Houston , Texas , USA;2. Centre for Mathematical Sciences , University of Cambridge , Cambridge , UK;3. Department of Mathematics , The University of Texas at Austin , Texas , USA |
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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 (amin {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. |
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Keywords: | Boltzmann equation Differential inequality Exponential moments Polynomial moments Povzner's estimates |
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