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      Short time diffusion properties of inhomogeneous kinetic equations with fractional collision kernel

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          Abstract

          We prove regularization properties in short time for inhomogeneous kinetic equations whose collision kernel behaves like a fractional power of the Laplacian in velocity. We treat a fractional Kolmogorov equation and the linearized Boltzmann equation without cutoff (for hard potentials).

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          Entropy Dissipation and Long-Range Interactions

          Marcelo Alexandre, L. Desvillettes, C Villani … (2000)
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            Regularizing effect and local existence for non-cutoff Boltzmann equation

            , , … (2009)
            The Boltzmann equation without Grad's angular cutoff assumption is believed to have regularizing effect on the solution because of the non-integrable angular singularity of the cross-section. However, even though so far this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo differential operators, we prove the regularizing effect in all (time, space and velocity) variables on solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and Maxwellian type decay in velocity variable, there exists a unique local solution with the same regularity, so that this solution enjoys the \(C^\infty\) regularity for positive time.
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              Global Classical Solutions of the Boltzmann Equation without Angular Cut-off

              Robert Strain, Philip Gressman (2010)
              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 \cite{gsNonCutA}. We more generally cover collision kernels with parameters \(s\in (0,1)\) and \(\gamma\) satisfying \(\gamma > -n\) in arbitrary dimensions \(\mathbb{T}^n \times \mathbb{R}^n\) with \(n\ge 2\). Moreover, we prove rapid convergence as predicted by the celebrated Boltzmann \(H\)-theorem. When \(\gamma \ge -2s\), we have exponential time decay to the Maxwellian equilibrium states. When \(\gamma <-2s\), 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 \(\gamma \ge -2s\), as conjectured in Mouhot-Strain \cite{MR2322149}. 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.
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                Publication date Created: 28 September 2017
                Article
                ArXiV ID: 1709.09943
                SO-VID: 08c4cafe-b7c4-4578-bf8e-6a58ebb25afc
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                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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