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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 inverse-power intermolecular potentials, r −(p−1) with p> 2, for initial perturbations of the Maxwellian equilibrium states, as ann ..."
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Cited by 44 (8 self)
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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 inverse-power 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 H-theorem. 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 Mouhot-Strain [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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Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space
"... In this paper we study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann 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 large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann 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 Vlasov-Poisson-Boltzmann 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
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.
Spectral and phase space analysis of the linearized non-cutoff Kac collision operator http://arxiv.org/abs/1111.0423
"... Abstract. The non-cutoff Kac operator is a kinetic model for the non-cutoff radially symmetric Boltzmann operator. For Maxwellian molecules, the linearization of the non-cutoff 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 non-cutoff Kac operator is a kinetic model for the non-cutoff radially symmetric Boltzmann operator. For Maxwellian molecules, the linearization of the non-cutoff 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 non-cutoff radially symmetric Boltzmann operator are also proven. 1.
Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators
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The Vlasov-Poisson-Boltzmann system without angular cutoff, preprint 2012
"... Abstract. This paper is concerned with the Vlasov-Poisson-Boltzmann system for plasma particles of two species in three space dimensions. The Boltzmann collision kernel is assumed to be angular non-cutoff with −3 < γ < −2s and 1/2 ≤ s < 1, where γ, s are two parameters describing the kineti ..."
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Cited by 4 (3 self)
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Abstract. This paper is concerned with the Vlasov-Poisson-Boltzmann system for plasma particles of two species in three space dimensions. The Boltzmann collision kernel is assumed to be angular non-cutoff with −3 < γ < −2s and 1/2 ≤ s < 1, where γ, s are two parameters describing the kinetic and angular singularities, respectively. We establish the global existence and convergence rates of classical solutions to the Cauchy problem when initial data is near Maxwellians. This extends the results in [10, 11] for the cutoff kernel with −2 ≤ γ ≤ 1 to the case −3 < γ < −2 as long as the angular singularity exists instead and is strong enough, i.e., s is close to 1. The proof is based on the time-weighted energy method building also upon the recent studies of the non cutoff Boltzmann equation in [13] and the
STABILITY OF THE NONRELATIVISTIC VLASOV-MAXWELL-BOLTZMANN SYSTEM FOR ANGULAR NON-CUTOFF POTENTIALS
"... Abstract. Although there recently have been extensive studies on the pertur-bation theory of the angular non-cutoff Boltzmann equation (cf. [4] and [17]), it remains mathematically unknown when there is a self-consistent Lorentz force coupled with the Maxwell equations in the nonrelativistic approxi ..."
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Cited by 2 (2 self)
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Abstract. Although there recently have been extensive studies on the pertur-bation theory of the angular non-cutoff Boltzmann equation (cf. [4] and [17]), it remains mathematically unknown when there is a self-consistent Lorentz force coupled with the Maxwell equations in the nonrelativistic approxima-tion. In the paper, for perturbative initial data with suitable regularity and integrability, we establish the large time stability of solutions to the Cauchy problem on the Vlasov-Maxwell-Boltzmann system with physical angular non-cutoff intermolecular collisions including the inverse power law potentials, and also obtain as a byproduct the convergence rates of solutions. The proof is based on a refined time-velocity weighted energy method with two key tech-nical parts: one is to introduce the exponentially weighted estimates into the non-cutoff Boltzmann operator and the other to design a delicate temporal en-ergy X(t)-norm to obtain its uniform bound. The result also extends the case of the hard sphere model considered by Guo (Invent. Math. 153(3): 593–630 (2003)) to the general collision potentials.
LOCAL EXISTENCE WITH MILD REGULARITY FOR THE BOLTZMANN EQUATION
"... Abstract. Without Grad’s angular cutoff assumption, the local existence of classical solutions to the Boltzmann equation is studied. There are two new improvements: the index of Sobolev spaces for the solution is related to the parameter of the angular singularity; moreover, we do not assume that th ..."
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Cited by 2 (0 self)
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Abstract. Without Grad’s angular cutoff assumption, the local existence of classical solutions to the Boltzmann equation is studied. There are two new improvements: the index of Sobolev spaces for the solution is related to the parameter of the angular singularity; moreover, we do not assume that the initial data is close to a global equilibrium. Using the energy method, one important step in the analysis is the study of fractional derivatives of the collision operator and related commutators. 1.
