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Optimal largetime behavior of the VlasovMaxwellBoltzmann system in the whole space
"... In this paper we study the largetime behavior of classical solutions to the twospecies VlasovMaxwellBoltzmann system in the whole space R³. The existence of global in time nearby Maxwellian solutions is known from [37] in 2006. However the asymptotic behavior of these solutions has been a chal ..."
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Cited by 29 (20 self)
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In this paper we study the largetime behavior of classical solutions to the twospecies VlasovMaxwellBoltzmann system in the whole space R³. The existence of global in time nearby Maxwellian solutions is known from [37] in 2006. However the asymptotic behavior of these solutions has been a challenging open problem. Buildingon ourprevious work[12]on timedecay for the simpler VlasovPoissonBoltzmann system, we prove that these solutions converge to the global Maxwellian with the optimal decay rate of O(t − 3 2 in L2 ξ (Lrx + 3
Optimal time decay of the non cutoff Boltzmann equation in the whole space
, 2010
"... Abstract. In this paper we study the largetime behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cutoff assumption in the whole space Rn x with n ≥ 3. We use the existence theory of global in time nearby Maxwellian solutions from [13 ..."
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Cited by 13 (4 self)
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Abstract. In this paper we study the largetime behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cutoff assumption in the whole space Rn x with n ≥ 3. We use the existence theory of global in time nearby Maxwellian solutions from [13,14]. It has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cutoff assumption [3, 29]. For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal largetime
Hypoelliptic estimates for a linear model of the Boltzmann equation without angular cutoff
"... Abstract. In this paper, we establish optimal hypoelliptic estimates for a class of kinetic equations, which are simplified linear models for the spatially inhomogeneous Boltzmann equation without angular cutoff. 1. ..."
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Cited by 11 (5 self)
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Abstract. In this paper, we establish optimal hypoelliptic estimates for a class of kinetic equations, which are simplified linear models for the spatially inhomogeneous Boltzmann equation without angular cutoff. 1.
ANISOTROPIC HYPOELLIPTIC ESTIMATES FOR LANDAUTYPE OPERATORS
"... Abstract. We establish global hypoelliptic estimates for linear Landautype operators. Linear Landautype equations are a class of inhomogeneous kinetic equations with anisotropic diffusion whose study is motivated by the linearization of the Landau equation near the Maxwellian distribution. By intr ..."
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Cited by 10 (3 self)
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Abstract. We establish global hypoelliptic estimates for linear Landautype operators. Linear Landautype equations are a class of inhomogeneous kinetic equations with anisotropic diffusion whose study is motivated by the linearization of the Landau equation near the Maxwellian distribution. By introducing a microlocal method by multiplier which can be adapted to various hypoelliptic kinetic equations, we establish for linear Landautype operators optimal global hypoelliptic estimates with loss of 4/3 derivatives in a Sobolev scale which is exactly related to the anisotropy of the diffusion. 1.
The VlasovPoissonBoltzmann system for soft potentials
 Math. Models Methods Appl. Sci
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The Boltzmann equation, Besov spaces, and optimal time decay rates in the whole space. Preprint 2012. See also arXiv:1206.0027
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Spectral and phase space analysis of the linearized noncutoff Kac collision operator http://arxiv.org/abs/1111.0423
"... Abstract. The noncutoff Kac operator is a kinetic model for the noncutoff radially symmetric Boltzmann operator. For Maxwellian molecules, the linearization of the noncutoff Kac operator around a Maxwellian distribution is shown to be a function of the harmonic oscillator, to be diagonal in the H ..."
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Cited by 7 (5 self)
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Abstract. The noncutoff Kac operator is a kinetic model for the noncutoff radially symmetric Boltzmann operator. For Maxwellian molecules, the linearization of the noncutoff Kac operator around a Maxwellian distribution is shown to be a function of the harmonic oscillator, to be diagonal in the Hermite basis and to be essentially a fractional power of the harmonic oscillator. This linearized operator is a pseudodifferential operator, and we provide a complete asymptotic expansion for its symbol in a class enjoying a nice symbolic calculus. Related results for the linearized noncutoff radially symmetric Boltzmann operator are also proven. 1.
Golobal solution and time decay of the VlasovPoissonLandau System in R3 x
 SIAM J. Math. Anal
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SMOOTHING EFFECT OF WEAK SOLUTIONS FOR THE SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION WITHOUT ANGULAR CUTOFF
, 2011
"... Abstract. In this paper, we consider the spatially homogeneous Boltzmann equation without angular cutoff. We prove that every L 1 weak solution to the Cauchy problem with finite moments of all order acquires the C ∞ regularity in the velocity variable for the positive time. ..."
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Cited by 5 (4 self)
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Abstract. In this paper, we consider the spatially homogeneous Boltzmann equation without angular cutoff. We prove that every L 1 weak solution to the Cauchy problem with finite moments of all order acquires the C ∞ regularity in the velocity variable for the positive time.