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Global Classical Solutions of the Boltzmann Equation with LongRange Interactions and Soft Potentials
"... Abstract. This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inversepower intermolecular potentials, r −(p−1) with p> 2, for initial perturbations of the Maxwellian equilibrium states, as ann ..."
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Abstract. This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inversepower intermolecular potentials, r −(p−1) with p> 2, for initial perturbations of the Maxwellian equilibrium states, as announced in [48]. We more generally cover collision kernels with parameters s ∈ (0,1) and γ satisfying γ> −n in arbitrary dimensions T n ×R n with n ≥ 2. Moreover, we prove rapid convergence as predicted by the celebrated Boltzmann Htheorem. When γ ≥ −2s, we have exponential time decay to the Maxwellian equilibrium states. When γ < −2s, our solutions decay polynomially fast in time with any rate. These results are completely constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when γ ≥ −2s, as conjectured in MouhotStrain [68]. It will be observed that this fundamental equation, derived by both Boltzmann and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the grazing collisions in the Boltzmann theory. Contents
Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential, to appear in Analysis and Applications
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Global existence and full regularity of the Boltzmann equation without angular cutoff
 Comm. Math. Phys
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Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft
 Potentials, Commun. Math. Phys
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The VlasovPoissonBoltzmann system for soft potentials
 Math. Models Methods Appl. Sci
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Commun. Math. Phys. Digital Object Identifier (DOI) 10.1007/s0022001011291 Communications in Mathematical Physics Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials
"... Abstract: In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in L ∞ ℓ. If the initial data are con ..."
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Abstract: In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in L ∞ ℓ. If the initial data are continuous then so is the corresponding solution. We work in the case of a spatially periodic box. Conditions on the collision kernel are generic in the sense of Dudyński and EkielJe˙zewska (Commun Math Phys 115(4):607–629, 1985); this resolves the open question of global existence for the soft potentials.