On the Boltzmann Equation for Fermi–Dirac Particles with Very Soft Potentials: Averaging Compactness of Weak Solutions |
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Authors: | Xuguang Lu |
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Affiliation: | (1) Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People's Republic of China |
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Abstract: | The paper considers macroscopic behavior of a Fermi–Dirac particle system. We prove the L 1-compactness of velocity averages of weak solutions of the Boltzmann equation for Fermi–Dirac particles in a periodic box with the collision kernel b(cos θ)|ρ−ρ *|γ, which corresponds to very soft potentials: −5 < γ ≤ −3 with a weak angular cutoff: ∫0 π b(cos θ)sin 3θ dθ < ∞. Our proof for the averaging compactness is based on the entropy inequality, Hausdorff–Young inequality, the L ∞-bounds of the solutions, and a specific property of the value-range of the exponent γ. Once such an averaging compactness is proven, the proof of the existence of weak solutions will be relatively easy. |
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Keywords: | Boltzmann equation Fermi– Dirac particles coulomb interaction weak angular cutoff averaging compactness |
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