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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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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 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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The VlasovPoissonBoltzmann system for soft potentials
 Math. Models Methods Appl. Sci
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Celebrating Cercignani’s conjecture for the Boltzmann equation
 Kinet. Relat. Models
"... Abstract. Cercignani’s conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann’s nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at ..."
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Cited by 7 (1 self)
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Abstract. Cercignani’s conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann’s nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s. This paper is dedicated to the memory of the late Carlo Cercignani, powerful mind and great scientist, one of the founders of the modern theory of the Boltzmann equation.
HILBERT’S 6TH PROBLEM: EXACT AND APPROXIMATE HYDRODYNAMIC MANIFOLDS FOR KINETIC EQUATIONS
"... Abstract. The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifolds were found analytically both as the ..."
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Cited by 5 (2 self)
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Abstract. The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifolds were found analytically both as the result of summation of the Chapman–Enskog asymptotic expansion and by the direct solution of the invariance equation. These model cases, comprising Grad’s moment systems, both linear and nonlinear, are studied in depth in order to gain understanding of what can be expected for the Boltzmann equation. Particularly, the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the shortwave limit and the viscosity modification at high divergence of the flow velocity are indicated as severe obstacles to the resolution of Hilbert’s 6th Problem. Furthermore, we review the derivation of the approximate hydrodynamic manifold for the Boltzmann equation using Newton’s iteration and avoiding smallness parameters, and compare this to the exact solutions. Additionally, we discuss the problem of projection of the
Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators
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Invariants and exponential rate of convergence to steady state in the renewal equation
 in &quot;Markov Processes and Related Fields (MPRF
"... Abstract We consider the renewal equation (also called McKendrickVonFoerster) equation that arises as a simple model for structured population dynamics. We use an entropy approach to prove the exponential convergence in long time to the steady state, after renormalization by a damping factor to com ..."
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Cited by 4 (0 self)
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Abstract We consider the renewal equation (also called McKendrickVonFoerster) equation that arises as a simple model for structured population dynamics. We use an entropy approach to prove the exponential convergence in long time to the steady state, after renormalization by a damping factor to compensate for the system growth. Our approach, by opposition with the original method of Feller based on Laplace transform, uses the direct variable. It uses new invariants of the equation, to which we systematically associate a condition for the exponential convergence.
The VlasovPoissonBoltzmann system without angular cutoff, preprint 2012
"... Abstract. This paper is concerned with the VlasovPoissonBoltzmann system for plasma particles of two species in three space dimensions. The Boltzmann collision kernel is assumed to be angular noncutoff 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 VlasovPoissonBoltzmann system for plasma particles of two species in three space dimensions. The Boltzmann collision kernel is assumed to be angular noncutoff 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 timeweighted energy method building also upon the recent studies of the non cutoff Boltzmann equation in [13] and the